Algebraic inclusions of Moufang polygons

Tom De Medts^{∗}

September 2, 2003

Abstract

An inclusion of a Moufang polygon into another is called algebraic if the algebraic structures which describe them can be chosen in such a way that the one is a substructure of the other. We show that an inclusion of Moufang n-gons is always algebraic if n ∈ {3, 6, 8}, but that this is not always true when n = 4. We classify the algebraic inclusions of Moufang quadrangles in the case where none of the root groups is 2-torsion, which corresponds to the fact that the characteristic of the underlying (skew) field is different from 2. Finally, we show that all full and ideal inclusions of Moufang quadrangles without 2-torsion root groups are algebraic.

MSC-2000 : primary: 51E12, secondary: 08A30, 20E42, 51E24

1 Introduction

A generalized polygon is a rank-2 incidence geometry the incidence graph of which has diameter n and girth 2n for some n ≥ 3 (and is then called a generalized n-gon). A generalized polygon is in fact the same as a rank-2 spherical building, and there is a vast literature on these objects. In many circumstances, one is interested in subpolygons of a given generalized polygon, for various reasons. To mention a few, they are used in characterizations of certain of these polygons, they can be used to discover or describe other interesting structures (such as spreads or ovoids), or they can be used in inductive arguments, for example to study embeddings of generalized polygons in projective spaces or other higher rank buildings.

A bit surprising, not too much has been written down on the study of subpolygons by itself. In the finite case, there are some results involving the order of the polygons; see, for example, [10, section 1.8]. The case of the classical compact connected polygons has been dealt with in [11].

In this paper, we will be interested in the case of the generalized polygons satisfying the Moufang condition. Although this condition looks rather restrictive, it is satisfied quite often, and in particular, all classical polygons belong to this class. Moreover, the Moufang polygons have been classified in [9] — but there is no hope to be able to classify all generalized polygons, since there exist free constructions, and even the finite case is still wide open. Two small pieces of the study of the inclusion of Moufang polygons have already been done before, namely the inclusion of Moufang octagons [5] and a study of the Moufang polygons which do not have full or ideal subpolygons [10, section 5.9]. It is also noteworthy that Moufang polygons (and spherical buildings of arbitrary rank in general) play an important role in algebraic group theory and related subjects, so our results might have consequences on the existence of subgroups of those groups.

∗The author is a Research Assistant of the Fund for Scientific Research - Flanders (Belgium)

(F.W.O.-Vlaanderen).

We will start by recalling some definitions and notations, and prepare the setup for our algebraic approach to the problem. Then we will be dealing with the cases of Moufang triangles, hexagons and octagons, which can be completely settled by taking a closer look at the proof of the classification of Moufang polygons in [9]. It goes without saying that we will have to rely heavily on this book. The case of Moufang quadrangles is significantly harder, and it turns out that the inclusion of Moufang polygons does not always translate nicely into the inclusion of the corresponding algebraic structures. However, in many cases, it does, and in particular in the case that the characteristic of the underlying (skew) field is not 2, we show that all inclusions are either algebraic or dual (see below for the exact definitions of these expressions). In the section which follows, we then classify all algebraic inclusions, with the only restriction that we do not allow the characteristic to be equal to 2 — a case which seems to be much harder (although many of our results can be extended to this case as well). In the last section, we describe all full and ideal subquadrangles of a given Moufang quadrangle, and we show that our list is complete.

2 Preliminaries

We start with some definitions.

Definition 2.1. A generalized n-gon is a connected bipartite graph with diameter n and girth 2n. A generalized polygon is a generalized n-gon for some finite n ≥ 2. A generalized polygon Γ is called thick if |Γ_{x}| ≥ 3 for all vertices x of Γ. A circuit of Γ of length 2n is called an apartment of Γ. A path of length n in Γ is called a root or a half-apartment of Γ.

Definition 2.2. If α = (v_{0}, …, v_{n}) is a root of a generalized n-gon Γ, then the group of all automorphisms of Γ which fix all the vertices of Γ_{v_{1}} ∪ ⋯ ∪ Γ_{v_{n−1}} is called a root group of Γ (corresponding to the root α) and is denoted by U_{α}. If U_{α} acts regularly on the set of apartments through α, then α is called a Moufang root. If all roots of Γ are Moufang roots, then Γ is called a Moufang n-gon.

From now on, we assume that Γ is a thick Moufang n-gon for some n ≥ 3, and we will fix an (arbitrary) apartment Σ which we label by the integers modulo 2n such that i + 1 ∈ Γ_{i} and i + 2 ≠ i for all i. We define U_{i} := U_{(i,i+1,...,i+n)} for all i, and we set U_{[i,j]} = ⟨U_{i}, U_{i+1}, …, U_{j}⟩ for all i ≤ j < i + n and U_{[i,i−1]} = 1 for all i.

The definitions 2.3–2.6 were introduced in [9]. We present them in a different but equivalent form.

Definition 2.3. Let Û_{[1,n]} be a group generated by non-trivial subgroups Û_{1}, …, Û_{n} for some n ≥ 3. The (n + 1)-tuple (Û_{[1,n]}, Û_{1}, …, Û_{n}) is called a root group sequence if there exists a Moufang n-gon Γ and a labeled apartment Σ = (0, …, 2n − 1) in Γ such that there exists an isomorphism from Û_{[1,n]} to U_{[1,n]} mapping Û_{i} to U_{i} for all i ∈ {1, …, n}. We will denote this root group sequence by Θ(Γ, Σ). The number n will be called the length of the root group sequence.

Definition 2.4. If Θ = (U_{[1,n]}, U_{1}, …, U_{n}) is a root group sequence, then (U_{[1,n]}, U_{n}, …, U_{1}) is also a root group sequence. It is called the opposite of Θ and is denoted by Θ^{op}.

Definition 2.5. Consider two root group sequences Θ = (U_{[1,n]}, U_{1}, …, U_{n}) and Θ′ = (U′_{[1,n]}, U′_{1}, …, U′_{n}). An isomorphism from Θ to Θ′ is an isomorphism from U_{[1,n]} to U′_{[1,n]} mapping U_{i} to U′_{i} for all i ∈ {1, …, n}. An anti-isomorphism from Θ to Θ′ is an isomorphism from Θ to Θ^{′ op}.

Definition 2.6. Let Θ = (U_{[1,n]}, U_{1}, …, U_{n}) be a root group sequence. For each i ∈ {1, …, n}, let U′_{i} be a non-trivial subgroup of U_{i}, and let U′_{[1,n]} denote the subgroup of U_{[1,n]} generated by U′_{1}, …, U′_{n}. If the n-tuple Θ′ = (U′_{[1,n]}, U′_{1}, …, U′_{n}) is again a root group sequence, then Θ′ will be called a subsequence of Θ.

Recently, the classification of Moufang polygons has been completed by J. Tits and R. Weiss in [9]. The following theorem is essential.

Theorem 2.7. Let Γ be an arbitrary Moufang n-gon. Then:

(i) n ∈ {3, 4, 6, 8}.

(ii) Let Σ = (0, …, 2n − 1) be an arbitrary apartment of Γ. Then up to isomorphism, Γ is uniquely determined by the isomorphism class of its root group sequence Θ(Γ, Σ) = (U_{[1,n]}, U_{1}, …, U_{n}). We denote this Moufang ngon by by Γ(Θ).

(iii) If Θ_{1} and Θ_{2} are two root group sequences such that Γ(Θ_{1}) ≅ Γ(Θ_{2}), then Θ_{1} and Θ_{2} are isomorphic or anti-isomorphic.

Proof. (i) See [9, (17.1)].

(ii) See [9, (7.6) and (7.7)].

(iii) Suppose that Θ_{1} = Θ(Γ_{1}, Σ_{1}) and Θ_{2} = Θ(Γ_{2}, Σ_{2}) for some Moufang n-gons Γ_{1} and Γ_{2} and some apartments Σ_{1} = (0, …, 2n − 1) and Σ_{2} = (0′, …, (2n − 1)′) of Γ_{1} and Γ_{2}, respectively. It follows from (ii) that Γ_{1} ≅ Γ(Θ_{1}) and Γ_{2} ≅ Γ(Θ_{2}), and hence Γ_{1} ≅ Γ_{2}. Let ϕ be an isomorphism from Γ_{1} to Γ_{2}, then ϕ maps Σ_{1} to some apartment ϕ(Σ_{1}) = (ϕ(0), …, ϕ(2n−1)) of Γ_{2}. By [9, (4.12)], there exists an automorphism ψ of Γ_{2} which maps ϕ(Σ_{1}) to Σ_{2} and maps the edge (ϕ(n), ϕ(n + 1)) to the edge (n′, (n + 1)′). So ψ ∘ ϕ maps Σ_{1} to Σ_{2}, and either it maps i to i for all i, in which case Θ_{1} and Θ_{2} are isomorphic, or it maps i to 2n + 1 − i for all i, in which case Θ_{1} and Θ_{2} are anti-isomorphic.

The following theorem defines the fundamental μ-maps, which play a very important role in the whole theory of Moufang polygons, in particular for the Moufang sets defined by opposite root groups in a Moufang polygon.

Theorem 2.8. For each i and each a_{i} ∈ U_{i}^{∗}, there exist a unique element μ(a_{i}) ∈ U_{i+n}^{∗}a_{i}U_{i+n}^{∗} such that (i − 1)^{μ(a_{i})} = i + 1 and (i + 1)^{μ(a_{i})} = i − 1. This element μ(a_{i}) fixes i and i + n and reflects Σ, and U_{j}^{μ(a_{i})} = U_{2i+n−j} for each a_{i} ∈ U_{i}^{∗} and each j.

Proof. See [9, (6.1)].

By the following theorem, the study of subpolygons of Moufang polygons is equivalent to the study of subsequences of root group sequences. Nevertheless, we will still use the polygons, since the μ-maps which we get from Theorem 2.8 will turn out to be very useful.

Theorem 2.9. (i) Let Γ_{2} be a Moufang n-gon and let Γ_{1} be a sub-n-gon of Γ_{2}. Then Γ_{1} is also a Moufang n-gon. If α is an arbitrary root of Γ_{1}, with corresponding root groups U_{α}^{(1)} of Γ_{1} and U_{α}^{(2)} of Γ_{2}, then U_{α}^{(1)} is a subgroup of U_{α}^{(2)}.

In particular, let Σ be an arbitrary labeled apartment of Γ_{1}, then Θ_{1} := Θ(Γ_{1}, Σ) is a subsequence of Θ_{2} := Θ(Γ_{2}, Σ).

(ii) Let Θ_{2} be a root group sequence and let Θ_{1} be a subsequence of Θ_{2}. Then Γ(Θ_{1}) is isomorphic to a subpolygon of Γ(Θ_{2}).

Proof. (i) The fact that Γ_{1} is again Moufang is well known; see, for example, [10, Lemma 5.2.2].

Consider an arbitrary root α of Γ_{1}, and its corresponding root groups U_{α}^{(1)} of Γ_{1} and U_{α}^{(2)} of Γ_{2}. Let Σ_{a} and Σ_{b} be two apartments of Γ_{1} through α. Then there is a unique element ϕ of U_{α}^{(2)} mapping Σ_{a} to Σ_{b}. Now consider the subgraph Δ := Γ_{1} ∩ ϕ(Γ_{1}). Since Γ_{1} and ϕ(Γ_{1}) have the apartment Σ_{b} in common, their intersection Δ is again a generalized n-gon (see, for example, [10, Proposition 1.8.4]). Since ϕ is a root elation, it fixes at least one pencil and at least one point row of Γ_{1}. It follows (see, for example, [10, Proposition 1.8.1]) that Δ is a full and ideal subpolygon of both Γ_{1} and ϕ(Γ_{1}), and hence (see, for example, [10, Proposition 1.8.2]) Δ, Γ_{1} and ϕ(Γ_{1}) coincide. We conclude that ϕ stabilizes Γ_{1}, and hence its restriction to Γ_{1} must be the unique element of U_{α}^{(1)} mapping Σ_{a} to Σ_{b}. Since this holds for every pair of apartments Σ_{a} and Σ_{b} of Γ_{1} through α, we have shown that every element of U_{α}^{(1)} is the restriction of a unique element of U_{α}^{(2)} to Γ_{1}. Hence U_{α}^{(1)} is a subgroup of U_{α}^{(2)}, for all roots α of Γ_{1}.

(ii) It follows readily from the construction in [9, (7.1) and (7.2)] that the vertex set X_{1} of Γ(Θ_{1}) can be canonically identified with a subset of the vertex set X_{2} of Γ(Θ_{2}), and that any two elements x, y ∈ X_{1} which are adjacent in Γ(Θ_{1}) are also adjacent in Γ(Θ_{2}). It follows that any two elements x, y ∈ X_{1} which have distance i in Γ(Θ_{1}) also have distance i in Γ(Θ_{2}), for all i ∈ {0, …, n}; in particular, two elements x, y ∈ X_{1} which are non-adjacent in Γ(Θ_{1}) are also non-adjacent in Γ(Θ_{2}). We conclude that Γ(Θ_{1}) is isomorphic to a subpolygon of Γ(Θ_{2}).

Theorem 2.10. If Θ = (U_{[1,n]}, U_{1}, …, U_{n}) is a root group sequence, then (i) [U_{i}, U_{j}] ≤ U_{[i+1,j−1]} for all 1 ≤ i < j ≤ n;

(ii) The product map from U_{1} × ⋯ × U_{n} to U_{[1,n]} is bijective.

Proof. See [9, (5.5) and (5.6)].

Definition 2.11. Let a_{i} ∈ U_{i} and a_{j} ∈ U_{j} with i + 2 ≤ j < i + n. For each k such that i < k < j, we set [a_{i}, a_{j}]_{k} = a_{k}, where a_{k} is the unique element of U_{k} appearing in the factorization of [a_{i}, a_{j}] ∈ U_{[i+1,j−1]}.

Lemma 2.12. Let Γ_{2} be a Moufang n-gon and let Γ_{1} be a sub-n-gon of Γ_{2}; let Σ be an arbitrary labeled apartment of Γ_{1}. Then the μ^{(1)}-maps defined by Theorem 2.8 with respect to Γ_{1} are the restriction of the μ^{(2)}-maps defined with respect to Γ_{2}, to the root groups of Γ_{1}.

Proof. Note that, by Theorem 2.9.(i), the root groups of Γ_{1} are indeed subgroups of the root groups of Γ_{2}, so the statement of this lemma makes sense.

But this same fact implies that every element μ^{(1)}(a_{i}) ∈ (U_{i+n}^{(1)})^{∗} ·a_{i} ·(U_{i+n}^{(1)})^{∗} with a_{i} ∈ (U_{i}^{(1)})^{∗} is also an element of (U_{i+n}^{(2)})^{∗}·a_{i}·(U_{i+n}^{(2)})^{∗}, and by the uniqueness of the μ^{(2)}-maps in Theorem 2.8, this element has to be equal to μ^{(2)}(a_{i}).

For each possible value of n, we will now use the appropriate algebraic structure to describe an arbitrary Moufang n-gon, and we will redo certain steps of the classification of Moufang n-gons, but for both the Moufang n-gon and its sub-n-gon simultaneously, and make some appropriate choices during the proof.

3 Subtriangles of Moufang triangles

We start with the study of all possible subtriangles of a given Moufang triangle. This is the easiest case, since Moufang triangles have a very simple description, as was already shown in 1933 (but in a slightly different form; see [1] or [3]) by R. Moufang (see [7]): Definition 3.1. Let (A, +, ·) be an arbitrary alternative division ring, and let U_{1}, U_{2} and U_{3} be three groups parametrized by (A, +) via some (group) isomorphisms x_{1}, x_{2} and x_{3}. Let U_{+} be the group generated by U_{1}, U_{2} and U_{3} with respect to the commutator relations

[U_{1}, U_{2}] = [U_{2}, U_{3}] = 1 ,
[x_{1}(s), x_{3}(t)] = x_{2}(s · t) ,

for all s, t ∈ A. Then Θ = (U_{+}, U_{1}, U_{2}, U_{3}) is a root group sequence of length 3; it is unique up to isomorphism (i.e., it does not depend on the choice of x_{1}, x_{2} and x_{3}), and will be denoted by Θ_{T} (A). We also say that Θ is parametrized by A via the isomorphisms x_{1}, x_{2} and x_{3}.

Theorem 3.2. Let Γ be an arbitrary Moufang triangle. Then there exists an alternative division ring (A, +, ·) such that Γ ≅ Γ(Θ_{T} (A)).

Proof. See [9, (17.2)].

Theorem 3.3. Let Γ_{1} and Γ_{2} be two Moufang triangles. Then Γ_{1} is isomorphic to a subtriangle of Γ_{2} if and only if there exists an alternative division ring Ã and a subring A of Ã such that Γ_{1} ≅ Γ(Θ_{T} (A)) and Γ_{2} ≅ Γ(Θ_{T} (Ã)).

Proof. (i) Suppose that Γ_{1} is a subtriangle of Γ_{2}; let Σ = (0, …, 5) be an arbitrary labeled apartment of Γ_{1}. We will write U_{i} and Ũ_{i} in place of U_{i}^{(1)} and U_{i}^{(2)}, respectively, to denote the root groups of Γ_{1} and Γ_{2} with respect to the labeled apartment Σ. By Theorem 2.9.(i), U_{i} ≤ Ũ_{i} for all i.

By [9, (19.4)], Ũ_{1} is abelian, and we choose an additive group Ã isomorphic to Ũ_{1} and an isomorphism t ↦ x_{1}(t) from Ã to Ũ_{1}. Let A := x_{1}^{−1}(U_{1}); then A is an additive group isomorphic to U_{1}. We now choose arbitrary elements e_{1} ∈ U_{1}^{∗} and e_{3} ∈ U_{3}^{∗}, and for every t ∈ Ã, we let

x_{2}(t) := x_{1}(t)^{μ(e_{3})} and x_{3}(t) := x_{2}(t)^{μ(e_{1})} ,

where μ is defined by Theorem 2.8 with respect to Γ_{2}. By Lemma 2.12 however, U_{2} = U_{1}^{μ(e_{3})} and U_{3} = U_{2}^{μ(e_{1})} as well, and hence U_{i} = x_{i}(A) for all i ∈ {1, 2, 3}.

Following [9, (19.6)], we now define a multiplication on Ã by defining uv = u · v to be the unique element of Ã such that

[x_{1}(u), x_{3}(v)] = x_{2}(uv) ,

for all u, v ∈ Ã. Since U_{i} = x_{i}(A), it follows that A is also closed under this multiplication. By [9, (19.7)], the left and right distributive laws hold in Ã, and therefore also in A. Now let 1 ∈ Ã^{∗} denote the element x_{1}^{−1}(e_{1}); then 1 ∈ A^{∗} as well. By [9, (19.9) and (19.13)], both Ã and A are alternative division rings with unit 1. In particular, A is a subring of Ã, and Γ_{1} ≅ Γ(Θ_{T} (A)) and Γ_{2} ≅ Γ(Θ_{T} (Ã)).

(ii) Let Θ_{2} = Θ_{T} (Ã) be the root group sequence parametrized by Ã via some isomorphisms x_{1}, x_{2} and x_{3}. Now let Θ_{1} = Θ_{T} (A) be the root group sequence parametrized by A via the restriction of these same isomorphisms x_{1}, x_{2} and x_{3} to A. Then Θ_{1} is a subsequence of Θ_{2}, and hence, by Theorem 2.9.(ii), Γ(Θ_{1}) is isomorphic to a subtriangle of Γ(Θ_{2}).

4 Subhexagons of Moufang hexagons

We postpone the case of Moufang quadrangles for a while, and we now consider subhexagons of a given Moufang hexagon. All Moufang hexagons can be parametrized by an anisotropic cubic norm structure, which is more often called an hexagonal system is this context (see [9, (15.15)]):

Definition 4.1. Let Ξ = (J, F, ♯) be an arbitrary hexagonal system with norm N, trace T, (Freudenthal) cross product × and unit 1 ∈ J^{∗}. Let U_{1}, U_{3} and U_{5} be three groups parametrized by J via some isomorphisms x_{1}, x_{3} and x_{5}, and let U_{2}, U_{4} and U_{6} be three groups parametrized by the additive group of F via some isomorphisms x_{2}, x_{4} and x_{6}. Let U_{+} be the group generated by U_{1}, …, U_{6} with respect to the commutator relations

[U_{1}, U_{2}] = [U_{1}, U_{4}] = [U_{2}, U_{3}] = [U_{2}, U_{4}] = [U_{2}, U_{5}] = 1 ,
[U_{3}, U_{4}] = [U_{3}, U_{6}] = [U_{4}, U_{5}] = [U_{4}, U_{6}] = [U_{5}, U_{6}] = 1 ,
[x_{1}(a), x_{3}(b)] = x_{2}(T (a, b)) ,
[x_{3}(a), x_{5}(b)] = x_{4}(T (a, b)) ,
[x_{1}(a), x_{5}(b)] = x_{2}(−T (a^{♯}, b))x_{3}(a × b)x_{4}(T (a, b^{♯})) ,
[x_{2}(t), x_{6}(u)] = x_{4}(tu) ,
[x_{1}(a), x_{6}(t)] = x_{2}(−tN(a))x_{3}(ta^{♯})x_{4}(t^{2}N(a))x_{5}(−ta) ,

for all a, b ∈ J and all t, u ∈ F. Then Θ = (U_{+}, U_{1}, U_{2}, U_{3}, U_{4}, U_{5}, U_{6}) is a root group sequence of length 6; it is unique up to isomorphism (i.e., it does not depend on the choice of the maps x_{i}), and will be denoted by Θ_{H}(Ξ). We also say that Θ is parametrized by Ξ via the isomorphisms x_{i}.

Theorem 4.2. Let Γ be an arbitrary Moufang hexagon. Then there exists an hexagonal system Ξ = (J, F, ♯) such that Γ ≅ Γ(Θ_{H}(Ξ)).

Proof. See [9, (17.5)].

Definition 4.3. Let Ξ̃ = (J̃, F̃,˜♯) be an hexagonal system. Then we say that an hexagonal system Ξ = (J, F, ♯) is a subsystem of Ξ̃ if F ⊆ F̃, J ⊆ J̃, the scalar multiplication F × J → J is the restriction of the scalar multiplication F̃ × J̃ → J̃, and ♯ is the restriction of ˜♯ to J.

Theorem 4.4. Let Γ_{1} and Γ_{2} be two Moufang hexagons. Then Γ_{1} is isomorphic to a subhexagon of Γ_{2} if and only if there exists an hexagonal system Ξ̃ and a subsystem Ξ of Ξ̃ such that Γ_{1} ≅ Γ(Θ_{H}(Ξ)) and Γ_{2} ≅ Γ(Θ_{H}(Ξ̃)).

Proof. (i) Suppose that Γ_{1} is a subhexagon of Γ_{2}; let Σ = (0, …, 9) be an arbitrary labeled apartment of Γ_{1}. We will write U_{i} and Ũ_{i} in place of U_{i}^{(1)} and U_{i}^{(2)}, respectively, to denote the root groups of Γ_{1} and Γ_{2} with respect to the labeled apartment Σ. By Theorem 2.9.(i), U_{i} ≤ Ũ_{i} for all i.

After relabeling the apartment Σ if necessary, we get, by [9, (29.11)], that Δ = (U_{2}U_{4}U_{6}, U_{2}, U_{4}, U_{6}) and Δ̃ = (Ũ_{2}Ũ_{4}Ũ_{6}, Ũ_{2}, Ũ_{4}, Ũ_{6}) are root group sequences (of length 3); in particular, Δ is a subsequence of Δ̃, and therefore Γ(Δ) is isomorphic to a subtriangle of Γ(Δ̃), by Theorem 2.9.(ii). By Theorem 3.3, there exists an alternative division ring F̃ and a subring F such that Γ(Δ) ≅ Γ(Θ_{T} (F)) and Γ(Δ̃) ≅ Γ(Θ_{T} (F̃)). By the argument following [9, (29.14)] however, F̃ is a commutative field. Using the fact that Θ_{T} (F)^{op} = Θ_{T} (F) for every commutative field F, it follows from Theorem 2.7.(iii) that Δ ≅ Θ_{T} (F) and Δ̃ ≅ Θ_{T} (F̃).

We now choose arbitrary elements e_{1} ∈ U_{1}^{∗} and e_{6} ∈ U_{6}^{∗}. By [9, (29.15)], there exist isomorphisms t ↦ x_{i}(t) from F̃ to Ũ_{i} for i ∈ {2, 4, 6} such that x_{6}(1) = e_{6}, x_{6}(t)^{μ(e_{1})} = x_{2}(t), x_{2}(t)^{μ(e_{1})} = x_{6}(−t) and [x_{2}(t), x_{6}(u)] = x_{4}(tu), for all t, u ∈ F̃. In particular, this last identity holds for all t, u ∈ F, and hence, by definition of the operator Θ_{T}, we have that Δ ≅ Θ_{T} (F) ≅ (x_{2}(F)x_{4}(F)x_{6}(F), x_{2}(F), x_{4}(F), x_{6}(F)). Hence we may assume that U_{i} = x_{i}(F) for all i ∈ {2, 4, 6}.

As in [9, (29.16)], we choose an additive group J̃ isomorphic to Ũ_{1} and an isomorphism a ↦ x_{1}(a) from J̃ to Ũ_{1}. Let x_{5}(a) := x_{1}(−a)^{μ(e_{6})} and x_{3}(a) := x_{5}(a)^{μ(e_{1})}, for all a ∈ J̃. Let J := x_{1}^{−1}(U_{1}). Then U_{i} = x_{i}(J) for i ∈ {1, 3, 5}.

Now let (t, a) ↦ ta be the map from F̃ × J̃ to J̃ defined so that

[x_{1}(a), x_{6}(t)^{−1}]_{5} = x_{5}(ta)  (1)

for all t ∈ F̃ and all a ∈ J̃, and let ♯ : J̃ → J̃ be the map defined by setting

[x_{1}(a), e_{6}]_{3} = x_{3}(a^{♯}) ,  (2)

for all a, b ∈ J̃. Then it is shown in [9, Chapter 29] that J̃ is a vector space over F̃ with scalar multiplication given by (t, a) ↦ ta, that Ξ̃ = (J̃, F̃, ♯) is an hexagonal system, and that Γ_{2} ≅ Γ(Θ_{H}(Ξ̃)). Since [x_{1}(J), x_{6}(F)]_{5} = [U_{1}, U_{6}]_{5} ⊆ U_{5}, it follows from (1) that F · J = J, and since [x_{1}(J), e_{6}]_{3} = [U_{1}, e_{6}]_{3} ⊆ U_{3}, it follows from (2) that J^{♯} ⊆ J. So by applying the same arguments, we can also conclude that Ξ = (J, F, ♯) is an hexagonal system, and that Γ_{1} ≅ Γ(Θ_{H}(Ξ)). Clearly, Ξ is a subsystem of Ξ̃.

(ii) Let Ξ = (J, F, ♯) and Ξ̃ = (J̃, F̃, ♯). Let Θ_{2} = Θ_{H}(Ξ̃) be the root group sequence parametrized by Ξ̃ via some isomorphisms x_{i}. Now let Θ_{1} = Θ_{H}(Ξ) be the root group sequence parametrized by Ξ via the restriction of these same isomorphisms x_{1}, x_{3} and x_{5} to J and the restriction of x_{2}, x_{4} and x_{6} to F. Then Θ_{1} is a subsequence of Θ_{2}, and hence, by Theorem 2.9.(ii), Γ(Θ_{1}) is isomorphic to a subhexagon of Γ(Θ_{2}).

5 Suboctagons of Moufang octagons

We now consider suboctagons of a given Moufang octagon. Although this has already been solved in [5, Theorem B], we also give a complete proof of this result, to illustrate that our approach also works for the case of Moufang octagons.

All Moufang octagons can be parametrized by a so-called octagonal set (see [9, (10.11)]):

Definition 5.1. Let (K, σ) be an arbitrary octagonal set, let U_{1}, U_{3}, U_{5} and U_{7} be four groups parametrized by the additive group of K via some isomorphisms x_{1}, x_{3}, x_{5} and x_{7}, and let U_{2}, U_{4}, U_{6} and U_{8} be four groups parametrized by K_{σ}^{(2)} via some isomorphisms x_{2}, x_{4}, x_{6} and x_{8}. Let U_{+} be the group generated by U_{1}, …, U_{8} with respect to certain commutator relations which can be found in [9, (16.9)]. Then Θ = (U_{+}, U_{1}, …, U_{8}) is a root group sequence of length 8; it is unique up to isomorphism (i.e., it does not depend on the choice of the maps x_{i}), and will be denoted by Θ_{O}(K, σ). We also say that Θ is parametrized by (K, σ) via the isomorphisms x_{i}.

Theorem 5.2. Let Γ be an arbitrary Moufang octagon. Then there exists an octagonal set (K, σ) such that Γ ≅ Γ(Θ_{O}(K, σ)).

Proof. See [9, (17.7)].

Definition 5.3. Let (K̃, σ̃) be an octagonal set. Then we say that an octagonal set (K, σ) is a subset of (K̃, σ̃) if K ⊆ K̃ and if σ is the restriction of σ̃ to K.

Theorem 5.4. Let Γ_{1} and Γ_{2} be two Moufang octagons. Then Γ_{1} is isomorphic to a suboctagon of Γ_{2} if and only if there exists an octagonal set (K̃, σ) and a subset (K, σ) of (K̃, σ) such that Γ_{1} ≅ Γ(Θ_{O}(K, σ)) and Γ_{2} ≅ Γ(Θ_{O}(K̃, σ)).

Proof. (i) Suppose that Γ_{1} is a suboctagon of Γ_{2}; let Σ = (0, …, 11) be an arbitrary labeled apartment of Γ_{1}. We will write U_{i} and Ũ_{i} in place of U_{i}^{(1)} and U_{i}^{(2)}, respectively, to denote the root groups of Γ_{1} and Γ_{2} with respect to the labeled apartment Σ. By Theorem 2.9.(i), U_{i} ≤ Ũ_{i} for all i.

After relabeling the apartment Σ if necessary, we get, by [9, (31.8)], that Δ = (U_{1}U_{3}U_{5}U_{7}, U_{1}, U_{3}, U_{5}, U_{7}) is the root group sequences of an indifferent Moufang quadrangle.

Let V_{i} := [U_{i−2}, U_{i+1}] and Ṽ_{i} := [Ũ_{i−2}, Ũ_{i+1}] for all even i. By [9, (31.16) and (31.29)], this definition of V_{i} and Ṽ_{i} coincides with the definition given in [9, (31.1)]. Clearly, V_{i} ≤ Ṽ_{i} for all even i.

We now choose arbitrary elements e_{1} ∈ U_{1}^{∗} and e_{8} ∈ V_{8}^{∗}. By [9, (31.24)], there exists an octagonal set (K̃, σ), and isomorphisms t ↦ x_{i}(t) from K̃ to Ũ_{i} for i ∈ {1, 3, 5, 7} such that x_{1}(1) = e_{1},

x_{i}(t)^{μ(e_{8})} = x_{8−i}(t)  (3)

for i ∈ {1, 3, 5, 7} and for all t ∈ K̃, and

[x_{1}(t), x_{7}(u)] = x_{3}(t^{σ}u)x_{5}(tu^{σ})  (4)

for all t, u ∈ K̃. Moreover, we let

x_{9}(t) := x_{1}(t)^{μ(e_{1})}  (5)

for all t ∈ K̃.

Now let K := x_{1}^{−1}(U_{1}); then (K, +) is an additive subgroup of (K̃, +) isomorphic to U_{1}. From (3) for i = 1, we get that x_{7}(K) = x_{1}(K)^{μ(e_{8})} = U_{1}^{μ(e_{8})} = U_{7}, and similarly, it follows from (5) that x_{9}(K) = U_{9}. It follows from (4) that x_{3}(K) = [e_{1}, x_{7}(K)]_{3} = [e_{1}, U_{7}]_{3} = U_{3}, and by (3) with i = 3, we get that x_{5}(K) = U_{5} as well.

By [9, (31.9.ii) and (31.32)], we have that μ(x_{1}(t)) = x_{9}(t^{−1})x_{1}(t)x_{9}(t^{−1}) for all t ∈ K̃. If we restrict this identity to K, then it follows from the fact that μ(a_{1}) ∈ U_{9}^{∗}a_{1}U_{9}^{∗} for all a_{1} ∈ U_{1}^{∗} and from Lemma 2.12 that K is closed under inverses.

Let x_{6}(t) := [x_{3}(t), e_{8}] for all t ∈ K̃. By the argument following [9, (31.25)], the map a_{3} ↦ [a_{3}, e_{8}] is an isomorphism from Ũ_{3} to Ṽ_{6}, and hence x_{6} is an isomorphism from K̃ to Ṽ_{6}. By the same argument, the map a_{3} ↦ [a_{3}, e_{8}] restricted to U_{3} is an isomorphism from U_{3} to V_{6}, and since U_{3} = x_{3}(K), the restriction of x_{6} to K is an isomorphism from K to V_{6}. If we now set x_{4}(t) := x_{6}(t)^{μ(e_{1})}, x_{2}(t) := x_{6}(t)^{μ(e_{8})} and x_{8}(t) := x_{2}(t)^{μ(e_{1})} for all t ∈ K̃, then Ṽ_{i} = x_{i}(K̃) and V_{i} = x_{i}(K) for all i ∈ {2, 4, 6, 8}.

By [9, (31.26.ii)], we have that [x_{1}(t), x_{6}(u)] = x_{4}(tu) for all t, u ∈ K̃. If we restrict this identity to t, u ∈ K, then it follows from the fact that [U_{1}, V_{6}] = V_{4} that K is closed under multiplication, and hence K is a subfield of K̃. Moreover, it follows from (4) that K^{σ} ⊆ K, and hence (K, σ) is a subset of (K̃, σ).

By the remainder of the classification result in [9, Chapter 31], we get that Γ_{1} ≅ Γ(Θ_{O}(K, σ)) and Γ_{2} ≅ Γ(Θ_{O}(K̃, σ)).

(ii) Let Θ_{2} = Θ_{O}(K̃, σ) be the root group sequence parametrized by (K̃, σ) via some isomorphisms x_{i}. Now let Θ_{1} = Θ_{O}(K, σ) be the root group sequence parametrized by (K, σ) via the restriction of these same isomorphisms x_{1}, x_{3}, x_{5} and x_{7} to K and the restriction of x_{2}, x_{4}, x_{6} and x_{8} to K_{σ}^{(2)}. Then Θ_{1} is a subsequence of Θ_{2}, and hence, by Theorem 2.9.(ii), Γ(Θ_{1}) is isomorphic to a suboctagon of Γ(Θ_{2}).

6 Subquadrangles of Moufang quadrangles

We now consider subquadrangles of a given Moufang quadrangle. In this case, the result is not as nice as in the other cases; in particular, it is not true that every inclusion of Moufang quadrangles can be described by the inclusion of the corresponding algebraic structures, as we will see in Theorem 6.6. The situation is not too bad, however, as will be illustrated by Lemma 6.15.

All Moufang quadrangles can be parametrized by a so-called quadrangular system (see [2]): Definition 6.1. Let Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) be an arbitrary quadrangular system with corresponding biadditive maps F and H; let U_{1} and U_{3} be two groups parametrized by W via some isomorphisms x_{1} and x_{3}, and let U_{2} and U_{4} be two groups parametrized by V via some isomorphisms x_{2} and x_{4}. Let U_{+} be the group generated by U_{1}, U_{2}, U_{3} and U_{4} with respect to the commutator relations

[U_{1}, U_{2}] = [U_{2}, U_{3}] = [U_{3}, U_{4}] = 1 ,
[x_{1}(w_{1}), x_{3}(w_{2})] = x_{2}(H(w_{1}, w_{2})) ,
[x_{2}(v_{1}), x_{4}(v_{2})] = x_{3}(F (v_{1}, v_{2})) ,
[x_{1}(w), x_{4}(v)] = x_{2}(vw)x_{3}(wv) ,

for all v, v_{1}, v_{2} ∈ V and all w, w_{1}, w_{2} ∈ W, where we have denoted the maps τ_{V} and τ_{W} by juxtaposition, i.e. vw := τ_{V} (v, w) and wv := τ_{W} (w, v) for all v ∈ V and all w ∈ W. Then Θ = (U_{+}, U_{1}, U_{2}, U_{3}, U_{4}) is a root group sequence of length 4; it is unique up to isomorphism (i.e., it does not depend on the choice of the maps x_{i}), and will be denoted by Θ_{Q}(Ω). We also say that Θ is parametrized by Ω via the isomorphisms x_{i}.

Theorem 6.2. Let Γ be an arbitrary Moufang quadrangle. Then there exists a quadrangular system Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) such that Γ ≅ Γ(Θ_{Q}(Ω)).

Proof. See [2, Section 5].

Definition 6.3. Let Γ be a Moufang quadrangle, and let Σ = (0, …, 7) be a labeled apartment of Γ. Let U_{0}, …, U_{7} be the root groups associated to Σ. Then we write V_{i} := [U_{i−1}, U_{i+1}] ≤ U_{i} for all i, and we let Y_{i} := C_{U_{i}}(U_{i−2}) ≤ U_{i} for each i. By [9, (21.20.i)], we have Y_{i} = C_{U_{i}}(U_{i+2}) as well.

Theorem 6.4. By relabeling the vertices of Σ by the transformation i ↦ 5 − i if necessary, we can assume that

(i) Y_{i} ≠ 1, [U_{i}, U_{i}] ≤ V_{i} ≤ Y_{i} ≤ Z(U_{i}) for all odd i;

(ii) U_{i} is abelian for all even i.

Proof. See [9, (21.28)].

Definition 6.5. Let Ω̃ = (Ṽ , W̃ , τṼ , τW̃ , ϵ̃, δ̃) be a quadrangular system. Then we say that a quadrangular system Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) is a subsystem of Ω̃ if V ⊆ Ṽ, W ⊆ W̃, ϵ = ϵ̃, δ = δ̃, and if τ_{V} and τ_{W} are the restriction of τṼ and τW̃ to V × W and W × V, respectively.

Theorem 6.6. Let Γ_{1} and Γ_{2} be two Moufang quadrangles.

(i) Suppose that Γ_{1} is a subquadrangle of Γ_{2}. Let Σ be an apartment of Γ_{1}, labeled in such a way that the statements of Theorem 6.4 hold for the root groups Ũ_{i} of Γ_{2}. If Y_{1} ∩ Ỹ_{1} ≠ 1, and if one of the conditions

(a) Ỹ_{4} = 1 ,

(b) Ỹ_{4} ≠ 1, Y_{4} = 1 and U_{4} ∩ Ỹ_{4} ≠ 1 ,

(c) Ỹ_{4} ≠ 1, Y_{4} ≠ 1 and Y_{4} ∩ Ỹ_{4} ≠ 1 ,

is satisfied, then there exists a quadrangular system Ω̃ and a subsystem Ω of Ω̃ such that Γ_{1} ≅ Γ(Θ_{Q}(Ω)) and Γ_{2} ≅ Γ(Θ_{Q}(Ω̃)).

(ii) If Γ_{1} ≅ Γ(Θ_{Q}(Ω)) and Γ_{2} ≅ Γ(Θ_{Q}(Ω̃)) for some quadrangular system Ω̃ and some subsystem Ω of Ω̃, then Γ_{1} is isomorphic to a subquadrangle of Γ_{2}.

Proof. (i) Suppose that Γ_{1} is a subquadrangle of Γ_{2}. Let Σ = (0, …, 7) be an apartment of Γ_{1}, labeled in such a way that the statements of Theorem 6.4 hold for the root groups of Γ_{2}. As before, we will write U_{i} and Ũ_{i} in place of U_{i}^{(1)} and U_{i}^{(2)}, respectively, to denote the root groups of Γ_{1} and Γ_{2} with respect to the labeled apartment Σ, and by Theorem 2.9.(i), U_{i} ≤ Ũ_{i} for all i.

We first show that the statements of Theorem 6.4 also hold for the root groups U_{i} of Γ_{1}. By applying this theorem on Γ_{1}, we see that either the statements hold for the given labeling, or they hold for the labeling transformed by the map i ↦ 5 − i. We may assume the latter. Then U_{i} is abelian for all odd i, and it is then obvious that [U_{i}, U_{i}] ≤ V_{i} and Y_{i} ≤ Z(U_{i}). By Definition 6.3, the statement V_{i} ≤ Y_{i} is equivalent to [[U_{i−1}, U_{i+1}], U_{i−2}] = 1; hence it follows from Ṽ_{i} ≤ Ỹ_{i} that V_{i} ≤ Y_{i} for all odd i. Finally, it follows from the assumption Y_{1} ∩ Ỹ_{1} ≠ 1 that Y_{i} ≠ 1 for all odd i, and it follows from the fact that Ũ_{i} is abelian for all even i that U_{i} is abelian for all even i. So the statements of Theorem 6.4 hold for the given labeling of the root groups U_{i}, after all.

By Theorem 6.4.(ii), Ũ_{4} is abelian, so choose a group (Ṽ , +) isomorphic to U_{4} and an isomorphism v ↦ x_{4}(v) from Ṽ to Ũ_{4}, and choose a (possibly non-abelian) group (W̃ , ⊞) isomorphic to U_{1} and an isomorphism w ↦ x_{1}(w) from W̃ to Ũ_{1}. Let V := x_{4}^{−1}(U_{4}) and let W := x_{1}^{−1}(U_{1}).

Since Y_{1}∩Ỹ_{1} ≠ 1, we can choose an element e_{1} ∈ Y_{1}^{∗}∩Ỹ_{1}^{∗}; let δ := x_{1}^{−1}(e_{1}). If we are in case (a) and Y_{4} = 1, then choose e_{4} ∈ U_{4}^{∗} arbitrarily; if Y_{4} ≠ 1, then choose e_{4} ∈ Y_{4}^{∗} arbitrarily. In both cases, e_{4} ∈ Ũ_{4}^{∗} as well. If we are in case (b), then choose e_{4} ∈ U_{4}^{∗} ∩ Ỹ_{4}^{∗}. Finally, if we are in case (c), then choose e_{4} ∈ Y_{4}^{∗} ∩ Ỹ_{4}^{∗}. Let ϵ := x_{1}^{−1}(e_{4}). Then δ and ϵ satisfy the assumptions which are required in [2, Section 5], for both Γ_{1} and Γ_{2}.

We now set x_{3}(w) := [x_{1}(w), e_{4}^{−1}]_{3} and x_{5}(w) := x_{1}(w)^{μ(e_{1})} for all w ∈ W̃ and x_{2}(v) := [e_{1}, x_{4}(v)^{−1}]_{2} and x_{0}(v) := x_{4}(v)^{μ(e_{4})} for all v ∈ Ṽ. Then Ũ_{i} = x_{i}(Ṽ) and U_{i} = x_{i}(V) for i ∈ {0, 2, 4}, and Ũ_{i} = x_{i}(W̃) and U_{i} = x_{i}(W) for i ∈ {1, 3, 5}.

As in [2, Section 5], we define a map τṼ from Ṽ × W̃ to Ṽ and a map τW̃ from W̃ × Ṽ to W̃, both of which are usually denoted by · or by juxtaposition, by setting

[x_{1}(w), x_{4}(v)^{−1}]_{2} = x_{2}(τṼ (v, w)) = x_{2}(vw) ,
[x_{1}(w), x_{4}(v)^{−1}]_{3} = x_{3}(τW̃ (w, v)) = x_{3}(wv) ,

for all w ∈ W̃ and all v ∈ Ṽ. It follows from the previous paragraph that τṼ (V, W) ⊆ V and τW̃ (W, V) ⊆ W. We denote these restrictions of τṼ and τW̃ by τ_{V} and τ_{W}, respectively.

By the remaining part of [2, Section 5], Ω̃ := (Ṽ , W̃ , τṼ , τW̃ , ϵ, δ) and Ω := (V, W, τ_{V} , τ_{W} , ϵ, δ) are quadrangular systems, Γ_{1} ≅ Γ(Θ_{Q}(Ω)) and Γ_{2} ≅ Γ(Θ_{Q}(Ω̃)). It is clear from the previous paragraph that Ω is a subsystem of Ω̃.

(ii) Let Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) and Ω̃ = (Ṽ , W̃ , τ_{V} , τ_{W} , ϵ, δ). Let Θ_{2} = Θ_{Q}(Ω̃) be the root group sequence parametrized by Ω̃ via some isomorphisms x_{i}. Now let Θ_{1} = Θ_{Q}(Ω) be the root group sequence parametrized by Ω via the restriction of these same isomorphisms x_{1} and x_{3} to W and the restriction of x_{2} and x_{4} to V. Then Θ_{1} is a subsequence of Θ_{2}, and hence, by Theorem 2.9.(ii), Γ(Θ_{1}) is isomorphic to a subquadrangle of Γ(Θ_{2}).

Definition 6.7. Let Γ_{1} and Γ_{2} be two Moufang quadrangles, and suppose that Γ_{1} is a subquadrangle of Γ_{2}. Let Σ be an apartment of Γ_{1}, labeled in such a way that the statements of Theorem 6.4 hold for the root groups Ũ_{i} of Γ_{2}. Suppose that the statements of Theorem 6.4 do not hold for the root groups U_{i} of Γ_{1}; then by this theorem, they do hold after the relabeling i ↦ 5 − i. Then we say that Γ_{1} is dually included in Γ_{2}. Note that this definition is independent of the choice of Σ.

If there exists a quadrangular system Ω̃ and a subsystem Ω of Ω̃ such that Γ_{1} ≅ Γ(Θ_{Q}(Ω)) and Γ_{2} ≅ Γ(Θ_{Q}(Ω̃)), then we say that Γ_{1} is algebraically included in Γ_{2}.

Remark 6.8. In section 8, we will give an example of an inclusion which is not algebraic but which is dual (8.1), as well as an example which is neither algebraic nor dual (8.2).

Definition 6.9. A quadrangular system Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) is called indifferent if F ≡ 0 and H ≡ 0, reduced if F ≢ 0 and H ≡ 0, co-reduced if F ≡ 0 and H ≢ 0 and wide if F ≢ 0 and H ≢ 0. A Moufang quadrangle is called indifferent, reduced or wide if it is parametrized by a quadrangular system which is indifferent, (co-)reduced or wide, respectively.

Remark 6.10. If Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) is a co-reduced quadrangular system, then Ω^{∗} := (W, V, τ_{W} , τ_{V} , δ, ϵ) is a reduced quadrangular system; see [2, Theorem 8.12]. Note that this can only occur if V and W are both 2-torsion groups, i.e. every element of V and W has order 1 or 2.

Lemma 6.11. (i) If δ ∈ W has order 2 (in particular, if W is a 2-torsion group), then V is a 2-torsion group.

(ii) If V contains an element of order 2, then V is a 2-torsion group.

Proof. From the defining axioms (Q_{9}) and (Q_{12}) in [2, Section 2], we get that v(δ ⊞ δ) = v + v for all v ∈ V.

(i) If δ has order 2, then v + v = v(δ ⊞ δ) = 0 for all v ∈ V.

(ii) Let c ∈ V^{∗} be an element for which c+c = 0, then c(δ ⊞δ) = 0, and hence δ ⊞ δ = 0, so by (i), V is a 2-torsion group.

Lemma 6.12. If Rad(F) ≠ 0, then V and W are 2-torsion groups.

Proof. See [2, Lemma 8.10].

The quadrangular systems have been classified; see [9, (17.4)] or [2]. It turns out that there are six different classes of quadrangular systems, which we will list now. We refer to [2] for more details about their definition.

Theorem 6.13. Let Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) be an arbitrary quadrangular system, and assume that Ω is not co-reduced. Then (at least) one of the following holds:

(i) Ω is indifferent, and Ω ≅ Ω_{D}(K, K_{0}, L_{0}) for some indifferent set (K, K_{0}, L_{0}). We say that Ω is of indifferent type.

(ii) Ω is reduced, and Ω ≅ Ω_{I}(K, K_{0}, σ) for some involutory set (K, K_{0}, σ). We say that Ω is of involutory type.

(iii) Ω is reduced, and Ω ≅ Ω_{Q}(K, V_{0}, q) for some anisotropic quadratic space (K, V_{0}, q). We say that Ω is of quadratic form type.

(iv) Ω is wide, and Ω ≅ Ω_{P} (K, K_{0}, σ, V_{0}, π) for some anisotropic pseudoquadratic space (K, K_{0}, σ, V_{0}, π). We say that Ω is of pseudo-quadratic form type.

(v) Ω is wide, and Ω ≅ Ω_{E}(K, V_{0}, q) for some quadratic space (K, V_{0}, q) of type E_{6}, E_{7} or E_{8}. We say that Ω is of type E_{6}, E_{7} or E_{8}.

(vi) Ω is wide, and Ω ≅ Ω_{F} (K, V_{0}, q) for some quadratic space (K, V_{0}, q) of type F_{4}. We say that Ω is of type F_{4}.

Moreover, if we are in case (i) or (vi), then V and W are 2-torsion groups.

Remark 6.14. It is obvious from the defining commutator relations in 6.1 that a (co-)reduced quadrangle can never be included in an indifferent quadrangle, and that a wide quadrangle cannot be included in a (co-)reduced quadrangle.

The conditions in Theorem 6.6.(i) look very restrictive, but in fact, they are satisfied quite often:

Lemma 6.15. Let Γ_{1} and Γ_{2} be two Moufang quadrangles, and suppose that Γ_{1} is a subquadrangle of Γ_{2}. Let Σ be an apartment of Γ_{1}, labeled in such a way that the statements of Theorem 6.4 hold for the root groups Ũ_{i} of Γ_{2}.

(i) If Ũ_{4} is not a 2-torsion group, then the inclusion of Γ_{1} in Γ_{2} is either algebraic or dual.

(ii) If Γ_{2} is of indifferent type or of involutory type, or if Γ_{2} ≅ Γ(Ω_{Q}(K, V_{0}, q)) for some regular quadratic form q, then the inclusion of Γ_{1} in Γ_{2} is algebraic.

Proof. (i) Suppose that the inclusion is not dual, so that the statements of Theorem 6.4 hold for the root groups U_{i} of Γ_{1} as well. Since Ũ_{4} is not 2-torsion, Lemma 6.11.(ii) implies that U_{4} is not 2-torsion either. We are in one of the cases (ii), (iii), (iv) or (v) of Theorem 6.13, and in each case, the fact that U_{4} ≅ V is not 2-torsion implies that the defining (skew) field K has characteristic different from 2. It is easily checked from their definition that Im(F) = Rad(H) ≠ 0 in each case, and hence V_{1} = Y_{1} ≠ 1. Similarly, Ṽ_{1} = Ỹ_{1} ≠ 1. Since V_{i} ≤ Ṽ_{i} for all i, it follows that Y_{1} ∩ Ỹ_{1} ≠ 1.

By Lemma 6.12, Rad(F̃) = 0, hence Ỹ_{4} = 1, so condition (a) of Theorem 6.6.(i) is satisfied. In particular, Γ_{1} is algebraically included in Γ_{2}.

(ii) If Γ_{2} is indifferent or reduced, then [Ũ_{1}, Ũ_{3}] = 1, and in particular, [U_{1}, U_{3}] = 1. However, this is equivalent to Ỹ_{1} = Ũ_{1} and Y_{1} = U_{1}, respectively, and in particular, Y_{1} ∩ Ỹ_{1} = U_{1} ≠ 1.

If Γ_{2} is indifferent, then the same argument applies to Ũ_{4}, and in particular Y_{4} ≠ 1, Ỹ_{4} ≠ 1 and Y_{4} ∩ Ỹ_{4} = U_{4} ≠ 1, so condition (c) of Theorem 6.6.(i) is satisfied.

If Γ_{2} is of involutory type but not indifferent, then Rad(F̂) = 0. If Γ_{2} ≅ Γ(Ω_{Q}(K, V_{0}, q)) for some quadratic form q with corresponding bilinear form f, then Rad(F̂) = Rad(f), so the condition on q to be regular implies that Rad(F̂) = 0 in this case as well. In both cases, Ỹ_{4} = 1, so condition (a) of Theorem 6.6.(i) is satisfied.

In all these cases, we can conclude by Theorem 6.6.(i) that Γ_{1} is algebraically included in Γ_{2}.

7 Algebraic inclusions of Moufang quadrangles

In this section, we will always assume that Γ_{1} and Γ_{2} are Moufang quadrangles such that Γ_{1} is algebraically included in Γ_{2}. Moreover, we will assume that none of the root groups is 2-torsion. The goal of this section is to classify these inclusions.

By Definition 6.7, there exists a quadrangular system Ω̃ = (Ṽ , W̃ , τṼ , τW̃ , ϵ, δ) and a subsystem Ω = (V, W, τṼ , τW̃ , ϵ, δ) of Ω̃ such that Γ_{1} ≅ Γ(Θ_{Q}(Ω)) and Γ_{2} ≅ Γ(Θ_{Q}(Ω̃)). Since the root groups are not 2-torsion, neither are the groups V, W, Ṽ and W̃. In particular, we are in one (or more than one) of the cases (ii), (iii), (iv) or (v) of Theorem 6.13.

Definition 7.1. A quadrangular system which is of involutory type but not of quadratic form type will be called of proper involutory type. An involutory set (K, K_{0}, σ) is proper if and only if σ ≠ 1 and K is generated by K_{0} as a ring. By [9, (21.10)], a quadrangular system Ω_{I}(K, K_{0}, σ) is of proper involutory type if and only if (K, K_{0}, σ) is proper.

Definition 7.2. A quadrangular system of pseudo-quadratic form type is not necessarily wide. If it is, then it is called of proper pseudo-quadratic form type.

Remark 7.3. If a quadrangular system Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) is of involutory type Ω = Ω_{I}(K, K_{0}, σ), and V and W are not 2-torsion, then char(K) ≠ 2; in particular, K_{0} = Fix_{K}(σ), and Ω is completely determined by K and σ. Hence we will denote the involutory set by (K, σ) in this case, and we will write Ω = Ω_{I}(K, σ).

By [9, (21.14)], (K, σ) is always proper, unless Fix_{K}(σ) is a commutative field F, and either K = F and σ = 1, or K is a separable quadratic extension over F (and then σ ∈ Gal(K/F)^{∗}), or K is a quaternion division algebra over F (and then σ is the standard involution of K/F).

Remark 7.4. If a quadrangular system Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) is of pseudoquadratic form type Ω = Ω_{P} (K, K_{0}, σ, V_{0}, π), and V and W are not 2-torsion, then char(K) ≠ 2; in particular, K_{0} = Fix_{K}(σ), and Ω is completely determined by K, σ, V_{0} and π. Hence we will write Ω = Ω_{P} (K, σ, V_{0}, π).

By [9, (21.16)], Ω_{P} (K, σ, V_{0}, π) with char(K) ≠ 2 is of proper pseudoquadratic form type if and only if V_{0} ≠ 0 and σ ≠ 1.

Using Theorem 6.13, we can now conclude that Ω and Ω̃ are of exactly one of the following types: proper involutory, quadratic form, proper pseudo-quadratic form, E_{6}, E_{7} or E_{8}. We have summarized the different combinations in Table 1, and we will consider each of the cases separately.

≤  Ω̃_{I}^{proper}  Ω̃_{Q}  Ω̃_{P}^{proper}  Ω̃_{E}
Ω_{I}^{proper}  7.5  7.10  7.8  7.11
Ω_{Q}  7.18  7.6  7.20 7.21
Ω_{P}^{proper} 6.14 6.14  7.9  7.23
Ω_{E}  6.14 6.14 7.22 7.23

Table 1: Algebraic inclusions of Moufang quadrangles

From now on, assume that Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) and Ω̃ = (Ṽ , W̃ , τṼ , τW̃ , ϵ̃, δ̃) are quadrangular systems with V, W, Ṽ and W̃ not 2-torsion. Then Ω is isomorphic to a subsystem of Ω̃ if and only if there exist group monomorphisms ϕ : V ↪ Ṽ and ψ : W ↪ W̃ such that

ϕ(ϵ) = ϵ̃,  (6)
ψ(δ) = δ̃,  (7)
ϕ(v)ψ(w) = ϕ(vw) ,  (8)
ψ(w)ϕ(v) = ψ(wv) ,  (9)

for all v ∈ V and all w ∈ W. In each case, the groups V, W, Ṽ and W̃ will be parametrized by a certain algebraic structure, and we will denote the isomorphisms by square brackets, as in [2].

Theorem 7.5. Let Ω ≅ Ω_{I}(K, σ) and Ω̃ ≅ Ω_{I}(K̃, σ̃) for some involutory sets (K, σ) and (K̃, σ̃).

(i) Suppose that (K, σ) is proper. Then Ω is isomorphic to a subsystem of Ω̃ if and only if there exists a field monomorphism α from K into K̃ such that α ∘ σ = σ̃ ∘ α.

(ii) Suppose that (K, σ) is not proper. Then Ω is isomorphic to a subsystem of Ω̃ if and only if there exists a field monomorphism or a field antimonomorphism α from K into K̃ such that α ∘ σ = σ̃ ∘ α.

Proof. Let F := Fix_{K}(σ) and let F̃ := FixK̃(σ̃). By the definition of the operator Ω_{I}, we have that V = [K], W = [F], Ṽ = [K̃] and W̃ = [F̃]. If (K, σ) is not proper, then we are in one of the three cases of Remark 7.3, so in particular F ≤ Z(K) and t^{σ}t = tt^{σ} for all t ∈ K; it follows from these two observations that tst^{σ} = t^{σ}st for all t ∈ K and all s ∈ F.

First assume that there exists a field monomorphism α from K into K̃ or — but only if (K, σ) is not proper — a field anti-monomorphism α from K into K̃, such that α ∘ σ = σ̃ ∘ α. We define a map ϕ from V = [K] to Ṽ = [K̃] and a map ψ from W = [F] to W̃ = [F̃] by setting ϕ[t] := [α(t)] and ψ[s] := [α(s)] for all t ∈ K and all s ∈ F. Since α is an (additive) monomorphism, so are ϕ and ψ. Moreover,

ϕ(ϵ) = ϕ[1] = [˜1] = ϵ̃,
ψ(δ) = ψ[1] = [˜1] = δ̃,

and if α is a field monomorphism, then

ϕ[t]ψ[s] = [α(t)][α(s)] = [α(s)α(t)] = [α(st)] = ϕ[st] = ϕ([t][s]) ,
ψ[s]ϕ[t] = [α(s)][α(t)] = [α(t)σ̃α(s)α(t)] = [α(t^{σ}st)] = ψ[t^{σ}st] = ψ([s][t]) ,

for all t ∈ K and all s ∈ F, whereas if α is a field anti-monomorphism and (K, σ) is not proper, then ts = st and tst^{σ} = t^{σ}st for all t ∈ K and all s ∈ F, and therefore

ϕ[t]ψ[s] = [α(t)][α(s)] = [α(s)α(t)] = [α(ts)] = [α(st)] = ϕ[st] = ϕ([t][s]) ,
ψ[s]ϕ[t] = [α(s)][α(t)] = [α(t)σ̃α(s)α(t)]
= [α(tst^{σ})] = [α(t^{σ}st)] = ψ[t^{σ}st] = ψ([s][t]) ,

for all t ∈ K and all s ∈ F. In both cases, we can conclude that Ω is isomorphic to a subsystem of Ω̃.

Now assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from K to K̃ and a map β from F to F̃ by setting ϕ[t] := [α(t)] and ψ[s] := [β(s)] for all t ∈ K and all s ∈ F. Since ϕ and ψ are group monomorphisms, α and β are additive monomorphisms as well. The conditions (6)–(9) translate into the following:

α(1) = ˜1 ,  (10)
β(1) = ˜1 ,  (11)
β(s)α(t) = α(st) ,  (12)
α(t)σ̃β(s)α(t) = β(t^{σ}st) ,  (13)

for all t ∈ K and all s ∈ F. If we set t = 1 in (12), then we get that β is the restriction of α to F. By (12) again, α(s)α(t) = α(st) for all t ∈ K and all s ∈ F. If we set s = 1 in (13), then we get that α(t)σ̃α(t) = α(t^{σ}t) for all t ∈ K. If we replace t by t + 1 and subtract the original identity, then it follows that α(t)σ̃ + α(t) = α(t^{σ} + t), and since α is additive, we get that α ∘ σ = σ̃ ∘ α.

If (K, σ) is proper, then K is generated (as a ring) by F, and it follows by induction on the identity α(s)α(t) = α(st) for all t ∈ K and all s ∈ F that α is multiplicative on K, hence it is a field monomorphism.

If (K, σ) is not proper, we have to proceed in a different way. It follows from (12) and (13) that α(t^{σ}s)α(t) = α(t^{σ}st) for all t ∈ K and all s ∈ F. If we set s = (t^{σ})^{−1}t^{−1} ∈ F, then it follows that α(t^{−1})α(t) = α(1) = ˜1, and hence α preserves inverses. It then follows from Hua’s identity

aba = a − (a^{−1} + (b^{−1} − a)^{−1})^{−1}

that α(aba) = α(a)α(b)α(a) for all a, b ∈ K, and hence α is a Jordan homomorphism. It follows from a result by Jacobson and Rickart [4] that α is a homomorphism or an anti-homomorphism. Since we already now that α is injective, the proof is finished.

Theorem 7.6. Let Ω ≅ Ω_{Q}(K, V_{0}, q) and Ω̃ ≅ Ω_{Q}(K̃, Ṽ_{0}, q̃) for some anisotropic quadratic spaces (K, V_{0}, q) and (K̃, Ṽ_{0}, q̃) with base points e and ẽ, respectively. Then Ω is isomorphic to a subsystem of Ω̃ if and only if there exists a vector space monomorphism (β, α) from (K, V_{0}) into (K̃, Ṽ_{0}) such that α(e) = ẽ and q̃(α(v)) = β(q(v)) for all v ∈ V_{0}.

Proof. By the definition of the operator Ω_{Q}, we have that V = [V_{0}], W = [K], Ṽ = [Ṽ_{0}] and W̃ = [K̃].

First assume that there exists a vector space monomorphism (β, α) from (K, V_{0}) into (K̃, Ṽ_{0}) such that α(e) = ẽ and q̃(α(v)) = β(q(v)) for all v ∈ V_{0}. We define a map ϕ from V = [V_{0}] to Ṽ = [Ṽ_{0}] and a map ψ from W = [K] to W̃ = [K̃] by setting ϕ[v] := [α(v)] and ψ[t] := [β(t)] for all v ∈ V_{0} and all t ∈ K.

Since α and β are additive monomorphisms, so are ϕ and ψ. Moreover,

ϕ(ϵ) = ϕ[e] = [ẽ] = ϵ̃,
ψ(δ) = ψ[1] = [˜1] = δ̃,
ϕ[v]ψ[t] = [α(v)][β(t)] = [β(t)α(v)] = [α(tv)] = ϕ[tv] = ϕ([v][t]) ,
ψ[t]ϕ[v] = [β(t)][α(v)] = [β(t)q̃(α(v))] = [β(t)β(q(v))] = ψ[tq(v)] = ψ([t][v]) ,

for all v ∈ V_{0} and all t ∈ K, and we can conclude that Ω is isomorphic to a subsystem of Ω̃.

Now assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from V_{0} to Ṽ_{0} and a map β from K to K̃ by setting ϕ[v] := [α(v)] and ψ[t] := [β(t)] for all t ∈ K and all s ∈ F. Since ϕ and ψ are group monomorphisms, α and β are additive monomorphisms as well. The conditions (6)–(9) translate into the following:

α(e) = ẽ,  (14)
β(1) = ˜1 ,  (15)
β(t)α(v) = α(tv) ,  (16)
β(t)q̃(α(v)) = β(tq(v)) ,  (17)

for all v ∈ V_{0} and all t ∈ K. It only remains to show that β is multiplicative; it will then follow from (16) that (β, α) is a vector space morphism, and the condition on the quadratic forms follows from (17) with t = 1. But by repeated use of (16), we get that β(s)β(t)α(e) = β(s)α(te) = α(ste) = β(st)α(e) for all s, t ∈ K, and hence β is multiplicative.

Remark 7.7. Since the Moufang quadrangles arising from anisotropic quadratic spaces (K, V_{0}, q) and (K, V_{0}, γq) are isomorphic, for every value of γ ∈ K^{∗}, the base points e and ẽ in Theorem 7.6 can be chosen arbitrarily (there is no restriction, since char(K) ≠ 2, and hence Rad(f) = 0). Therefore the condition α(e) = ẽ is not really a restriction.

Theorem 7.8. Let Ω ≅ Ω_{I}(K, σ) and Ω̃ ≅ Ω_{P} (K̃, σ̃, Ṽ_{0}, π̃) for some involutory set (K, σ) and some anisotropic pseudo-quadratic space (K̃, σ̃, Ṽ_{0}, π̃). Then Ω is isomorphic to a subsystem of Ω̃ if and only if Ω is already isomorphic to a subsystem of Ω_{I}(K̃, σ̃).

Proof. Let F := Fix_{K}(σ) and let F̃ := FixK̃(σ̃). Denote the skew-hermitian form corresponding to Ω̃ by h̃. By the definition of the operators Ω_{I} and Ω_{P}, we have that V = [K], W = [F], Ṽ = [K̃] and W̃ = [T̃], where (T̃, ⊞) is the group with underlying set {(a, t) ∈ Ṽ_{0} × K̃ | π̃(a) − t ∈ F̃}, and with group action (a, t) ⊞ (b, s) = (a + b, t + s + h̃(b, a)) for all (a, t), (b, s) ∈ T̃.

First assume that Ω is isomorphic to a subsystem of Ω_{I}(K̃, σ̃). Since Ω_{I}(K̃, σ̃) is obviously a subsystem of Ω̃, it then follows that Ω is isomorphic to a subsystem of Ω̃.

So assume now that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from K to K̃, a map β from F to Ṽ_{0} and a map γ from F to K̃ by setting ϕ[t] := [α(t)] and ψ[s] := [β(s), γ(s)] ∈ [T̃] for all t ∈ K and all s ∈ F. Since ϕ and ψ are group morphisms, α and β are additive morphisms as well (but note that it does not follow that γ is additive). The conditions (7) and (9) imply the following:

β(1) = 0 ,  (18)
β(s)α(t) = β(t^{σ}st) ,  (19)

for all t ∈ K and all s ∈ F. We only need to show that β = 0, since it will then follow that ψ(F) ≤ (0, F̃), and hence Ω is in fact isomorphic to a subsystem of Ω_{I}(K̃, σ̃). If we set s = 1 in (19), then we get that β(t^{σ}t) = 0 for all t ∈ K. Replacing t by t + 1 and subtracting the original equation, we get that β(t + t^{σ}) = 0 for all t ∈ K. But since char(K) ≠ 2, this implies that β(s) = β((s/2) + (s/2)^{σ}) = 0 for all s ∈ F, and we are done.

Theorem 7.9. Let Ω ≅ Ω_{P} (K, σ, V_{0}, π) and Ω̃ ≅ Ω_{P} (K̃, σ̃, Ṽ_{0}, π̃) for some proper anisotropic pseudo-quadratic spaces (K, σ, V_{0}, π) and (K̃, σ̃, Ṽ_{0}, π̃). Denote the skew-hermitian forms corresponding to Ω and Ω̃ by h and h̃, respectively. Then Ω is isomorphic to a subsystem of Ω̃ if and only if there exists a vector space monomorphism (β, α) from (K, V_{0}) into (K̃, Ṽ_{0}) such that β ∘ σ = σ̃ ∘ β and β(h(a, b)) = h̃(α(a), α(b)) for all a, b ∈ V_{0}.

Proof. Let F := Fix_{K}(σ) and let F̃ := FixK̃(σ̃). By the definition of the operator Ω_{P}, we have that V = [K], W = [T], Ṽ = [K̃] and W̃ = [T̃], where the groups (T, ⊞) and (T̃, ⊞) are defined as in the previous theorem.

First assume that there exists a vector space monomorphism (β, α) from (K, V_{0}) into (K̃, Ṽ_{0}) such that β ∘ σ = σ̃ ∘ β and β(h(a, b)) = h̃(α(a), α(b)) for all a, b ∈ V_{0}. We define a map ϕ from V = [K] to Ṽ = [K̃] and a map ψ from W = [T] to W̃ = [T̃] by setting ϕ[s] := [β(s)] and ψ[a, t] := [α(a), β(t)] for all (a, t) ∈ T and all s ∈ K. Since α and β are additive monomorphisms, so are ϕ and ψ, because of the condition that β(h(a, b)) = h̃(α(a), α(b)) for all a, b ∈ V_{0}. Moreover,

ϕ(ϵ) = ϕ[1] = [˜1] = ϵ̃,
ψ(δ) = ψ[0, 1] = [0, ˜1] = δ̃,
ϕ[s]ψ[a, t] = [β(s)][α(a), β(t)] = [β(t)β(s)] = [β(ts)] = ϕ[ts] = ϕ([s][a, t]) ,
ψ[a, t]ϕ[s] = [α(a), β(t)][β(s)] = [α(a)β(s), β(s)σ̃β(t)β(s)]
= [α(as), β(s^{σ}ts)] = ψ[as, s^{σ}ts] = ψ([a, t][s]) ,

for all s ∈ K and all (a, t) ∈ T, and we can conclude that Ω is isomorphic to a subsystem of Ω̃.

Now assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map β from K to K̃, a map α from T to Ṽ_{0} and a map γ from T to K̃ by setting ϕ[t] := [β(t)] and ψ[a, t] := [α(a, t), γ(a, t)] ∈ [T̃] for all t ∈ K and all (a, t) ∈ T. If we restrict ψ to [0, F], then we are back in the situation of Theorem 7.8, and therefore β is a field monomorphism from K into K̃ such that β ∘ σ = σ̃ ∘ β; moreover, α(0, t) = 0 and γ(0, t) = β(t) for all t ∈ F.

Define α(a) := α(a, π(a)) for all a ∈ V_{0}. Since ψ is a group morphism, so is α : T → Ṽ_{0}, and hence

α(a, t) = α((a, π(a)) ⊞ (0, t − π(a)))
= α(a, π(a)) + α(0, t − π(a))
= α(a, π(a)) = α(a)

for all (a, t) ∈ T. Again using the fact that ψ is a group morphism, we have that

γ(a + b, t + s + h(b, a)) = γ(a, t) + γ(b, s) + h̃(α(b), α(a))  (20)

for all (a, t), (b, s) ∈ T. By condition (8), we have that γ(a, t)β(s) = β(ts) for all s ∈ K and all (a, t) ∈ T. Since β is multiplicative, we get that γ(a, t) = β(t) for all (a, t) ∈ T. If we apply this on equation (20), then we get, using the fact that β is additive, that β(h(b, a)) = h̃(α(b), α(a)) for all a, b ∈ V_{0}. It follows from (9) that α(a)β(t) = α(at) for all a ∈ V_{0} and all t ∈ K, so (β, α) is a vector space morphism.

It only remains to show that α : V_{0} → Ṽ_{0} is injective. So suppose that α(a) = α(b) for some a, b ∈ V_{0}. Note that (a, π(a)) and (b, π(b)) are contained in T. Since ψ[T] ≤ [T̃], we have that [α(a), β(π(a))] = ψ[a, π(a)] ∈ [T̃], and hence π̃(α(a)) − β(π(a)) ∈ F̃, and similarly π̃(α(b)) − β(π(b)) ∈ F̃. Since α(a) = α(b), this implies that β(π(b)−π(a)) ∈ F̃. It follows that β(π(b)−π(a)) is fixed under σ̃, and hence π(b) − π(a) is fixed under σ since β ∘ σ = σ̃ ∘ β and since β is injective. So π(b) − π(a) ∈ F, and hence (a, π(b)) ∈ T. But then

ψ[a, π(b)] = [α(a), β(π(b))] = [α(b), β(π(b))] = ψ[b, π(b)] ,

and since ψ is injective, it follows that a = b. We conclude that α is injective, and we are done.

Theorem 7.10. Let Ω ≅ Ω_{I}(K, σ) and Ω̃ ≅ Ω_{Q}(K̃, Ṽ_{0}, q̃) for some proper involutory set (K, σ) and some anisotropic quadratic space (K̃, Ṽ_{0}, q̃) with base point ẽ. Then Ω cannot be isomorphic to a subsystem of Ω̃.

Proof. Let F := Fix_{K}(σ). By the definition of the operators Ω_{I} and Ω_{Q}, we have that V = [K], W = [F], Ṽ = [Ṽ_{0}] and W̃ = [K̃].

Assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from K to Ṽ_{0} and a map β from F to K̃ by setting ϕ[t] := [α(t)] and ψ[s] := [β(s)] for all t ∈ K and all s ∈ F. Since ϕ and ψ are group monomorphisms, α and β are additive monomorphisms as well. The conditions (6)–(9) translate into the following:

α(1) = ẽ,  (21)
β(1) = ˜1 ,  (22)
β(s)α(t) = α(st) ,  (23)
β(s)q̃(α(t)) = β(t^{σ}st) ,  (24)

for all t ∈ K and all s ∈ F. If we set t = 1 in (23), then we get that α(s) = β(s)ẽ for all s ∈ F. By induction using (23) again, α(s_{1} ⋯ s_{n}) = β(s_{1}) ⋯ β(s_{n})ẽ for all s_{1}, …, s_{n} ∈ F. Since (K, σ) is proper, K is generated (as a ring) by F, and it follows that β can be extended to a field morphism β̂ : K → K̃ such that α(t) = β̂(t)ẽ. Since α is injective, so is β̂. It then follows from (24) that β̂(s)β̂(t)^{2} = β̂(t^{σ})β̂(s)β̂(t) for all t ∈ K and all s ∈ F, and hence β̂(t^{σ}) = β̂(t) for all t ∈ K. Since β̂ is injective, it follows that t^{σ} = t for all t ∈ K, which contradicts the properness of (K, σ). Hence Ω cannot be isomorphic to a subsystem of Ω̃.

Theorem 7.11. Let Ω ≅ Ω_{I}(K, σ) and Ω̃ ≅ Ω_{E}(K̃, Ṽ_{0}, q̃) for some proper involutory set (K, σ) and some quadratic space (K̃, Ṽ_{0}, q̃) of type E_{6}, E_{7} or E_{8}. Then Ω cannot be isomorphic to a subsystem of Ω̃.

Proof. Let F := Fix_{K}(σ). By the definition of the operators Ω_{I} and Ω_{E}, we have that V = [K], W = [F], Ṽ = [Ṽ_{0}] and W̃ = [S̃], where (S̃, ⊞) is the (non-abelian) group with underlying set X_{0} × K as defined in [9, (16.6)]. (See [9] for more details; we only mention that X_{0} is a certain vector space over K and that the group operation ⊞ is additive on the X_{0}-component.)

Assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map β from K to Ṽ_{0}, a map α from F to X̃_{0} and a map γ from F to K̃ by setting ϕ[t] := [β(t)] and ψ[s] := [α(s), γ(s)] ∈ [S̃] for all t ∈ K and all s ∈ F. Since ϕ and ψ are group morphisms, α and β are additive morphisms as well, By (7), ψ[1] = [0, 1], so in particular α(1) = 0. Condition (9) implies that α(t^{σ}st) = α(s)β(t) for all t ∈ K and all s ∈ F. If we set s = 1, then it follows that α(t^{σ}t) = 0 for all t ∈ K. Replacing t by t + 1 and subtracting the original equation, we obtain α(t + t^{σ}) = 0 for all t ∈ K, and hence α(F) = 0 since char(K) ≠ 2. It follows that ψ(W) ≤ [0, K], so in fact Ω is isomorphic to a subsystem of Ω_{Q}(K̃, Ṽ_{0}, q̃). But since Ω is of proper involutory type, this contradicts Theorem 7.10, so Ω cannot be isomorphic to a subsystem of Ω̃.

The next case we will deal with is by far the most interesting. We will first recall some definitions and facts from the theory of quadratic forms.

Definition 7.12. Let (K, V_{0}, q) be an anisotropic regular quadratic space (where K has arbitrary characteristic). Then we define the Clifford algebra C(V_{0}, q) := T (V_{0})/I(V_{0}, q), where T (V_{0}) is the tensor algebra of the vector space V_{0}, i.e. T (V_{0}) := K ⊕ V_{0} ⊕ (V_{0} ⊗ V_{0}) ⊕ (V_{0} ⊗ V_{0} ⊗ V_{0}) ⊕ ⋯, and I(V_{0}, q) is the ideal ⟨u ⊗ u − q(u) · 1 | u ∈ V_{0}⟩ of T (V_{0}). The multiplication in C(V_{0}, q) is usually denoted by juxtaposition, so in particular uu = q(u) ∈ K for all u ∈ V_{0}. The even Clifford algebra C_{0}(V_{0}, q) is the subalgebra of C(V_{0}, q) generated by the set {uv | u, v ∈ V_{0}}. The Clifford algebra and the even Clifford algebra admit a canonical involution τ : v_{1}v_{2} ⋯ v_{k} ↦ v_{k}v_{k−1} … v_{1} for all v_{1}, …, v_{k} ∈ V_{0}. If dim_{K} V_{0} = n, then dim_{K} C(V_{0}, q) = 2^{n} and dim_{K} C_{0}(V_{0}, q) = 2^{n−1}. Both the Clifford algebra and the even Clifford algebra are either simple, or the direct sum of two isomorphic simple algebras.

Definition 7.13. Let (K, V_{0}, q) be an anisotropic regular quadratic space (where K has arbitrary characteristic) with base point e ∈ V_{0}^{∗}; denote the bilinear form corresponding to q by f. Let \overline{v} := f(e, v)e − v for all v ∈ V_{0}. Then we define the Clifford algebra with base point C(V_{0}, q, e) := T (V_{0})/I(V_{0}, q, e), where I(V_{0}, q, e) is the ideal ⟨e − 1, u ⊗ \overline{u} − q(u) · 1 | u ∈ V_{0}⟩ of T (V_{0}). The multiplication in C(V_{0}, q, e) will also be denoted by juxtaposition, so in particular e = 1 and u\overline{u} = q(u) ∈ K for all u ∈ V_{0}. The Clifford algebra with base point admits a canonical involution τ_{e} : v_{1}v_{2} ⋯ v_{k} ↦ \overline{v_{k}} \overline{v_{k−1}} … \overline{v_{1}} for all v_{1}, …, v_{k} ∈ V_{0}. If dim_{K} V_{0} = n, then dim_{K} C(V_{0}, q, e) = 2^{n−1}.

Lemma 7.14. Let (K, V_{0}, q) be an anisotropic regular quadratic space with base point e ∈ V_{0}^{∗}. Then C_{0}(V_{0}, q) ≅ C(V_{0}, q, e). More precisely, there is a K-linear isomorphism χ : C(V_{0}, q, e) → C_{0}(V_{0}, q) such that χ(u) = eu for all u ∈ V_{0}. Moreover, χ ∘ τ_{e} = τ ∘ χ.

Proof. See, for example, [9, (12.51)], except for the last statement, which can be checked by a straightforward calculation.

The following definition will become clear in Theorem 7.18:

Definition 7.15. Let (K, V_{0}, q) be an anisotropic regular quadratic space with char(K) ≠ 2. Then we will say that (K, V_{0}, q) is involutoric if one of the following two conditions is satisfied:

(i) C_{0}(V_{0}, q) ≅ D for some division algebra D;

(ii) dim_{K} V_{0} ≡ 0 (mod 4) and C_{0}(V_{0}, q) ≅ D ⊕ D for some division algebra D.

In case (i), let τ be the canonical involution of C_{0}(V_{0}, q); in case (ii), it follows from [6, (8.4)] that the canonical involution of C_{0}(V_{0}, q) maps each of the two components two itself, and hence induces an involution τ on D. In both cases, we call Ω_{I}^{env}(K, V_{0}, q) := Ω_{I}(D, τ) the enveloping quadrangular system of involutory type of Ω_{Q}(K, V_{0}, q).

Remark 7.16. Let (K, V_{0}, q) be an involutoric quadratic space. Then Ω_{I}^{env}(K, V_{0}, q) is of proper involutory type, except if dim_{K} V_{0} ≤ 3, or if dim_{K} V_{0} = 4 and q is the norm form of a quaternion division algebra.

Lemma 7.17. Let (K, V_{0}, q) be an involutoric quadratic space with base point e, and let C := C(V, q, e). If C ≅ D for some division algebra D, then let π_{1} be the identity map from C to D. If C ≅ D ⊕ D for some division algebra D, then let π_{1} be the projection from C onto the first D-component. In both cases, the restriction of π_{1} to V_{0} is injective.

Proof. This is obvious if C ≅ D, so assume that C ≅ D ⊕ D. Since π_{1} is additive, it suffices to show that π_{1}(v) = 0 for some v ∈ V_{0} implies that v = 0.

So assume that v ∈ V_{0} is such that π_{1}(v) = 0. Then π_{1}(q(v)e) = π_{1}(v\overline{v}) = π_{1}(v)π_{1}(\overline{v}) = 0, but since π_{1}(e) = 1, this implies that q(v) = 0 and hence v = 0 since q is anisotropic.

Theorem 7.18. Let Ω ≅ Ω_{Q}(K, V_{0}, q) and Ω̃ ≅ Ω_{I}(K̃, σ̃) for some anisotropic quadratic space (K, V_{0}, q) with base point e and some involutory set (K̃, σ̃). Then Ω is isomorphic to a subsystem of Ω̃ if and only if (K, V_{0}, q) is involutoric and Ω_{I}^{env}(K, V_{0}, q) is isomorphic to a subsystem of Ω̃. In particular, Ω is isomorphic to a subsystem of Ω_{I}^{env}(K, V_{0}, q) itself.

Proof. Let F̃ := FixK̃(σ̃). By the definition of the operators Ω_{Q} and Ω_{I}, we have that V = [V_{0}], W = [K], Ṽ = [K̃] and W̃ = [F̃].

First assume that (K, V_{0}, q) is involutoric, and that Ω_{I}^{env}(K, V_{0}, q) is isomorphic to a subsystem of Ω̃. Without loss of generality, we may assume that Ω̃ = Ω_{I}^{env}(K, V_{0}, q). By Lemma 7.14, we may consider K̃ = D and F̃ = Fix_{K}(τ_{e}) as subalgebras of C := C(V_{0}, q, e). Let π_{1} : C → D be the morphism defined in Lemma 7.17; then π_{1}(V_{0}) ≤ K̃ and π_{1}(K) = π_{1}(Ke) ≤ F̃ since τ_{e}(e) = e. We define a map ϕ from V = [V_{0}] to Ṽ = [K̃] and a map ψ from W = [K] to W̃ = [F̃] by setting ϕ[v] := [π_{1}(v)] and ψ[t] := [π_{1}(t)] for all v ∈ V_{0} and all t ∈ K. Then it follows from Lemma 7.17 that ϕ and ψ are group monomorphisms. Moreover,

ϕ(ϵ) = ϕ[e] = [π_{1}(e)] = [1] = ϵ̃,
ψ(δ) = ψ[1] = [π_{1}(1)] = [1] = δ̃,
ϕ[v]ψ[t] = [π_{1}(v)][π_{1}(t)] = [π_{1}(t)π_{1}(v)] = [π_{1}(tv)] = ϕ[tv] = ϕ([v][t]) ,
ψ[t]ϕ[v] = [π_{1}(t)][π_{1}(v)] = [π_{1}(v)^{τ_{e}}π_{1}(t)π_{1}(v)]
= [π_{1}(\overline{v}tv)] = [π_{1}(tq(v))] = ψ[tq(v)] = ψ([t][v]) ,

for all v ∈ V_{0} and all t ∈ K, and we can conclude that Ω is isomorphic to a subsystem of Ω̃.

Now assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from V_{0} to K̃ and a map β from K to F̃ by setting ϕ[v] := [α(v)] and ψ[t] := [β(t)] for all v ∈ V_{0} and all t ∈ K. Since ϕ and ψ are group monomorphisms, α and β are additive monomorphisms as well. The conditions (6)–(9) translate into the following:

α(e) = ˜1 ,  (25)
β(1) = ˜1 ,  (26)
β(t)α(v) = α(tv) ,  (27)
α(v)σ̃β(t)α(v) = β(tq(v)) ,  (28)

for all v ∈ V_{0} and all t ∈ K. By repeated use of (27), we get that β(s)β(t)α(e) = β(s)α(te) = α(ste) = β(st)α(e) for all s, t ∈ K, hence β is multiplicative; it follows that β(K) is a commutative field which is contained in F̃. By (28) with t = 1, we get that

α(v)σ̃α(v) = β(q(v))  (29)

for all v ∈ V_{0}. Replacing v by v + e and subtracting the original equation yields α(v) + α(v)σ̃ = β(f(e, v)) and hence, by (27),

α(\overline{v}) = α(f(e, v)e − v) = β(f(e, v))α(e) − α(v) = α(v)σ̃  (30)

for all v ∈ V_{0}. It follows from (29) and (30) that α(\overline{v})α(v) = β(q(v)), and since α(e) = β(1) by (25) and (26), it follows that there exists an algebra morphism α̂ : C := C(V_{0}, q, e) → K̃ such that β is the restriction of α̂ to K and α is the restriction of α̂ to V_{0}.

Suppose first that C is simple. Since α and β are injective, the kernel of α̂ cannot be equal to C, hence it has to be trivial, and therefore α̂ is an algebra monomorphism. It follows that C is a division algebra. We will write C = D and γ = α̂ in this case, and we let τ be the standard involution τ_{e} of C.

Now suppose that C is the direct sum of two isomorphic simple algebras, say C = D ⊕ D. By the structure theory of Clifford algebras (see, for example, [6, (8.2)]), this can only occur if dim_{K} V_{0} is even. Again, the kernel of α̂ cannot be equal to C. It cannot be trivial, either, since C has zero divisors but K̃ does not. So it has to be equal to one of the two direct summands, say 0 ⊕ D. But then the restriction of α̂ to the first direct summand is injective, and hence the induced map from D to K̃ — which we will denote by γ — is an algebra monomorphism, so D is a division algebra in this case as well. By (30), \overline{v} ∈ ker α̂ if and only if v ∈ ker α̂. But if we would have dim_{K} V_{0} ≡ 2 (mod 4), then it would follow from [6, (8.4)] that the standard involution of the even Clifford algebra switches the two direct components, and using Lemma 7.14, we would obtain a contradiction. Hence dim_{K} V_{0} ≡ 0 (mod 4) in this case. Now we let τ be the restriction of the standard involution τ_{e} of C to its first component D.

So we have shown that (K, V_{0}, q) is involutoric. It remains to show that Ω_{I}^{env}(K, V_{0}, q) = Ω_{I}(D, τ) is isomorphic to a subsystem of Ω̃ = Ω_{I}(K̃, σ̃). But γ is an injective map from D into K̃, and it follows from (30) that γ ∘ τ = σ̃ ∘ γ. By Theorem 7.5, this finishes the proof of this theorem.

We have now reduced the case Ω_{Q} ≤ Ω_{I} to the case Ω_{I} ≤ Ω_{I} which we have already considered in Theorem 7.5.

Remark 7.19. The condition on a quadratic space to be involutoric looks very restrictive, and in fact, it is. Nevertheless, involutoric quadratic spaces exist in any dimension. The following example was communicated by J.-P. Tignol, and is a slight modification of the appendix of [8].

Let K be a field with char(K) ≠ 2. Suppose that Q_{1}, …, Q_{n} are quaternion algebras over K such that A := Q_{1} ⊗ ⋯ ⊗ Q_{n} is a division algebra, and denote by i_{1}, j_{1}, …, i_{n}, j_{n} the usual generators of the quaternion algebras Q_{1}, …, Q_{n}; moreover let k_{ℓ} := i_{ℓ}j_{ℓ} for every ℓ ∈ {1, …, n}. Consider the elements u_{ℓ} := k_{1} ⋯ k_{ℓ−1}i_{ℓ} and v_{ℓ} := k_{1} ⋯ k_{ℓ−1}j_{ℓ} for every ℓ ∈ {1, …, n − 1}, and let w := k_{1} ⋯ k_{n}. These elements pairwise anticommute and are square-central; denote their squares by a_{1}, b_{1}, …, a_{n}, b_{n}, c.

We can map the Clifford algebra of the 2n-dimensional quadratic form q = ⟨a_{1}, b_{1}, …, a_{n}, b_{n}⟩ to A by carrying the basis elements of the quadratic space to the elements u_{1}, v_{1}, …, u_{n}, v_{n}. This gives us an algebra homomorphism C(q) → A, which has to be injective since C(q) is simple, and has to be surjective by dimension count, and hence C(q) ≅ A. Since A is division, q is anisotropic.

If the quaternion algebras Q_{1}, …, Q_{n} are chosen in such a way that disc(q) ∉ K^{2}, then it follows that C_{0}(q) is also a division algebra, which is central over the discriminant extension field of q.

Similarly, we can consider the (2n + 1)-dimensional anisotropic quadratic form q′ = ⟨a_{1}, b_{1}, …, a_{n}, b_{n}, c⟩, and one can check that C_{0}(q′) ≅ A.

Theorem 7.20. Let Ω ≅ Ω_{Q}(K, V_{0}, q) and Ω̃ ≅ Ω_{P} (K̃, σ̃, Ṽ_{0}, π̃) for some anisotropic quadratic space (K, V_{0}, q) and some anisotropic pseudo-quadratic space (K̃, σ̃, Ṽ_{0}, π̃). Then Ω is isomorphic to a subsystem of Ω̃ if and only if Ω is already isomorphic to a subsystem of Ω_{I}(K̃, σ̃).

Proof. Let F̃ := FixK̃(σ̃). Denote the bilinear form corresponding to Ω by f. By the definition of the operators Ω_{Q} and Ω_{P}, we have that V = [V_{0}], W = [K], Ṽ = [K̃] and W̃ = [T̃], where the group (T, ⊞) is defined as in Theorem 7.8.

Assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from V_{0} to K̃, a map β from K to Ṽ_{0} and a map γ from K to K̃ by setting ϕ[v] := [α(v)] and ψ[t] := [β(t), γ(t)] ∈ [T̃] for all v ∈ V_{0} and all t ∈ K. Since ϕ and ψ are group morphisms, α and β are additive morphisms as well. The conditions (7) and (9) imply the following:

β(1) = 0 ,  (31)
β(t)α(v) = β(tq(v)) ,  (32)

for all v ∈ V_{0} and all t ∈ K. If we set t = 1 in (32), then we get that β(q(v)) = 0 for all v ∈ V_{0}. Linearizing this identity gives us that β(f(u, v)) = 0 for all u, v ∈ V_{0}. Since char(K) ≠ 2, f ≠ 0, and hence f is surjective, so β(K) = 0. It follows that ψ[K] ≤ [0, F̃], and hence Ω is in fact isomorphic to a subsystem of Ω_{I}(K̃, σ̃).

Theorem 7.21. Let Ω ≅ Ω_{Q}(K, V_{0}, q) and Ω̃ ≅ Ω_{E}(K̃, Ṽ_{0}, q̃) for some anisotropic quadratic space (K, V_{0}, q) and some anisotropic quadratic space (K̃, Ṽ_{0}, q̃) of type E_{6}, E_{7} or E_{8}. Then Ω is isomorphic to a subsystem of Ω̃ if and only if Ω is already isomorphic to a subsystem of Ω_{Q}(K̃, Ṽ_{0}, q̃).

Proof. The proof of this theorem is very similar to the proof of Theorem 7.20. Denote the bilinear form corresponding to Ω by f. By the definition of the operators Ω_{Q} and Ω_{E}, we have that V = [V_{0}], W = [K], Ṽ = [Ṽ_{0}] and W̃ = [S̃], where (S̃, ⊞) is the (non-abelian) group with underlying set X_{0} × K as defined in [9, (16.6)], as in Theorem 7.11.

Assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map α from V_{0} to Ṽ_{0}, a map β from K to X̃_{0} and a map γ from K to K̃ by setting ϕ[v] := [α(v)] and ψ[t] := [β(t), γ(t)] ∈ [S̃] for all v ∈ V_{0} and all t ∈ K. Since ϕ and ψ are group morphisms, α and β are additive morphisms as well. The conditions (7) and (9) imply the following:

β(1) = 0 ,  (33)
β(t)α(v) = β(tq(v)) ,  (34)

for all v ∈ V_{0} and all t ∈ K. If we set t = 1 in (34), then we get that β(q(v)) = 0 for all v ∈ V_{0}. Again, it follows that β = 0, and hence ψ[K] ≤ [0, K̃]. Therefore Ω is in fact isomorphic to a subsystem of Ω_{Q}(K̃, Ṽ_{0}, q̃).

Theorem 7.22. Let Ω ≅ Ω_{E}(K, V_{0}, q) and Ω̃ ≅ Ω_{P} (K̃, σ̃, Ṽ_{0}, π̃) for some anisotropic quadratic space (K, V_{0}, q) of type E_{6}, E_{7} or E_{8}, and some proper anisotropic pseudo-quadratic space (K̃, σ̃, Ṽ_{0}, π̃). Then Ω cannot be isomorphic to a subsystem of Ω̃.

Proof. Assume that Ω is isomorphic to a subsystem of Ω̃. Then in particular, Ω_{Q}(K, V_{0}, q) is isomorphic to a subsystem of Ω̃. By Theorem 7.20, this implies that Ω_{Q}(K, V_{0}, q) is isomorphic to a subsystem of Ω_{I}(K̃, σ̃). It thus follows from Theorem 7.18 that (K, V_{0}, q) has to be involutoric. But by [9, (12.43)], C_{0}(q) ≅ Mat_{4}(E) if q is of type E_{6}, C_{0}(q) ≅ Mat_{4}(D) ⊕ Mat_{4}(D) if q is of type E_{7}, and C_{0}(q) ≅ Mat_{32}(K) ⊕ Mat_{32}(K) if q is of type E_{8}. In all three cases, we obtain a contradiction, and hence Ω cannot be isomorphic to a subsystem of Ω̃.

The remaining two cases are very similar to each other, and we consider them together. The group S, the map (a, v) ↦ av from X_{0} × V_{0} → X_{0}, and the maps h, θ and g (and similar objects for Ω̃) are as in [9, (16.6)].

Theorem 7.23. (i) Let Ω ≅ Ω_{P} (K, σ, X_{0}, π) and Ω̃ ≅ Ω_{E}(K̃, Ṽ_{0}, q̃) for some proper anisotropic pseudo-quadratic space (K, σ, V_{0}, π) and some quadratic space (K̃, Ṽ_{0}, q̃) of type E_{6}, E_{7} or E_{8} with base point ẽ. Then Ω is isomorphic to a subsystem of Ω̃ if and only if (K, σ) is not proper, with Ω ≅ Ω_{Q}(F, K, q), and there exists a field monomorphism γ : F ↪ K̃ and γ-vector space monomorphisms β : K ↪ Ṽ_{0} and α : X_{0} ↪ X̃_{0} such that

β(1) = ẽ,
α(av) = α(a)β(v) ,
γ(q(v)) = q̃(β(v)) ,
h̃(α(a), α(b)) = β(h(a, b)) ,

for all a, b ∈ X_{0} and all v ∈ V_{0}.

(ii) Let Ω ≅ Ω_{E}(K, V_{0}, q) and Ω̃ ≅ Ω_{E}(K̃, Ṽ_{0}, q̃) for some quadratic spaces (K, V_{0}, q) and (K̃, Ṽ_{0}, q̃) of type E_{6}, E_{7} or E_{8}, with base points e and ẽ, respectively. Then Ω is isomorphic to a subsystem of Ω̃ if and only if there exists a field monomorphism γ : K ↪ K̃ and γ-vector space monomorphisms β : V_{0} ↪ Ṽ_{0} and α : X_{0} ↪ X̃_{0} such that

β(e) = ẽ,
α(av) = α(a)β(v) ,
γ(q(v)) = q̃(β(v)) ,
h̃(α(a), α(b)) = β(h(a, b)) ,

for all a, b ∈ X_{0} and all v ∈ V_{0}.

Proof. We start by showing that if Ω ≅ Ω_{P} (K, σ, X_{0}, π) is isomorphic to a subsystem of Ω̃ ≅ Ω_{E}(K̃, Ṽ_{0}, q̃), then the choice of the involutory set (K, σ) is very limited. So suppose that Ω ≤ Ω̃, then also Ω_{I}(K, σ) is isomorphic to a subsystem of Ω̃. Then Theorem 7.11 implies that (K, σ) is not proper. Also, since the pseudo-quadratic space (K, σ, X_{0}, π) is proper, we have σ ≠ 1. It follows from Remark 7.3 that Ω ≅ Ω_{Q}(F, K, q), where F is a commutative field, and K is either a separable quadratic extension field over F with norm q, or a quaternion division algebra over F with norm q. In these two cases, we can reparametrize the quadrangular system Ω in exactly the same way as we do for the exceptional quadrangular systems of type E_{6}, E_{7} and E_{8} (by defining V_{0} := K and S being the group with underlying set X_{0} × F defined as usual), using the isomorphism T → S : (a, t) ↦ (a, t − π(a)) for all (a, t) ∈ T; see also [9, (26.44)]. Note that the map (a, v) ↦ av from X_{0} × V_{0} = K → X_{0} and the map h : X_{0} × X_{0} → V_{0} = K do not change under this isomorphism.

Only to avoid confusion in the notation, we will assume from now on that we are in case (ii), even though the proof of case (i) is now completely identical.

First assume that there exists a field monomorphism γ : K ↪ K̃ and γvector space monomorphisms β : V_{0} ↪ Ṽ_{0} and α : X_{0} ↪ X̃_{0} such that

β(e) = ẽ,  (35)
α(av) = α(a)β(v) ,  (36)
γ(q(v)) = q̃(β(v)) ,  (37)
h̃(α(a), α(b)) = β(h(a, b)) ,  (38)

for all a, b ∈ X_{0} and all v ∈ V_{0}. Since g(a, b) = f(h(b, a), e)/2 for all a, b ∈ X_{0} by [9, (13.26)], and similarly for g̃, it follows from (35), (37) and (38) that g(α(a), α(b)) = γ(g(a, b)) for all a, b ∈ X_{0}. We define a map ϕ from V = [V_{0}] to Ṽ = [Ṽ_{0}] and a map ψ from W = [S] to W̃ = [S̃] by setting ϕ[v] := [β(v)] and ψ[a, t] := [α(a), γ(t)] for all v ∈ V_{0} and all (a, t) ∈ S. Since α and β are additive monomorphisms, so are ϕ and ψ, because of the condition that γ(g(a, b)) = g̃(α(a), α(b)) for all a, b ∈ X_{0}. Since θ(a, v) = h(a, av)/2 for all a ∈ X_{0} and all v ∈ V_{0} by [9, (13.28)], it follows from (36) and (38) that θ̃(α(a), β(v)) = β(θ(a, v)) for all a ∈ X_{0} and all v ∈ V_{0}, and since β is a γ-semilinear vector space isomorphism, we also have β(tv) = γ(t)β(v) for all t ∈ K and all v ∈ V_{0}. Using (35), (36) and (37), we thus get that

ϕ(ϵ) = ϕ[e] = [β(e)] = [ẽ] = ϵ̃,
ψ(δ) = ψ[0, 1] = [α(0), γ(1)] = [0, ˜1] = δ̃,
ϕ[v]ψ[a, t] = [β(v)][α(a), γ(t)] = [θ̃(α(a), β(v)) + γ(t)β(v)]
= [β(θ(a, v) + tv)] = ϕ[θ(a, v) + tv] = ϕ([v][a, t]) ,
ψ[a, t]ϕ[v] = [α(a), γ(t)][β(v)] = [α(a)β(v), γ(t)q̃(β(v))]
= [α(av), γ(tq(v))] = ψ[av, tq(v)] = ψ([a, t][v]) ,

for all v ∈ V_{0} and all (a, t) ∈ S, and we can conclude that Ω is isomorphic to a subsystem of Ω̃.

Now assume that Ω is isomorphic to a subsystem of Ω̃, with corresponding group monomorphisms ϕ and ψ satisfying (6)–(9). We define a map β from V_{0} to Ṽ_{0}, a map α from S to X̃_{0} and a map γ from S to K̃ by setting ϕ[t] := [β(t)] and ψ[a, t] := [α(a, t), γ(a, t)] ∈ [S̃] for all t ∈ K and all (a, t) ∈ S. Moreover, let α(a) := α(a, 0) and γ(t) := γ(0, t) for all a ∈ X_{0} and all t ∈ K. By (6), β(e) = ẽ. If we restrict ψ to [0, K], then we are back in the situation of Theorem 7.21, and therefore γ : K → K̃ is a field monomorphism and β : V_{0} → Ṽ_{0} is a γ−semilinear vector space monomorphism, such that q̃(β(v)) = γ(q(v)) for all v ∈ V_{0}; moreover, α(0, t) = 0 for all t ∈ F. Since ψ is additive,

α(a + b, t + s + g(a, b)) = α(a, t) + α(b, s) ,  (39)
γ(a + b, t + s + g(a, b)) = γ(a, t) + γ(b, s) + g̃(α(a, t), α(b, s)) ,  (40)

for all (a, t), (b, s) ∈ X_{0}. If we set t = 0 and b = 0 in (39), then we get that α(a, s) = α(a, 0) + α(0, s) for all (a, s) ∈ X_{0}, and hence α(a, s) = α(a) for all (a, s) ∈ X_{0}. Similarly, γ(a, s) = γ(a, 0)+γ(s) for all (a, s) ∈ X_{0}. If we substitute this last identity in (40), then we get that

γ(a + b, 0) + γ(g(a, b)) = γ(a, 0) + γ(b, 0) + g̃(α(a), α(b))  (41)

for all a, b ∈ X_{0}. Remember that char(K) ≠ 2. If we interchange a and b in (41), and add the result to (41), then we get, using the fact that g and g̃ are anti-symmetric by [9, (13.47.i)], that 2γ(a + b, 0) = 2(γ(a, 0) + γ(b, 0)) for all a, b ∈ X_{0}, and hence γ : S → K is additive in the X_{0}-component. The identities (8) and (9) translate into the following:

θ̃(α(a), β(v)) + γ(a, t)β(v) = β(θ(a, v) + tv) ,  (42)
α(a)β(v) = α(av) ,  (43)
γ(a, t)q̃(β(v)) = γ(av, tq(v)) ,  (44)

for all v ∈ V_{0} and all (a, t) ∈ S. If we substitute 2v for v and 0 for t in (44), then we get that γ(a, 0)q̃(2β(v)) = γ(2av, 0) for all a ∈ X_{0} and all v ∈ V_{0}. Using the fact that γ : S → K is additive in the X_{0}-component, it follows that 4γ(a, 0)q̃(β(v)) = 2γ(av, 0) = 2γ(a, 0)q̃(β(v)), and if we choose v ≠ 0, then q̃(β(v)) ≠ 0 since β is injective and q̃ is anisotropic; hence γ(a, 0) = 0 for all a ∈ X_{0}, and therefore γ(a, t) = γ(t) for all (a, t) ∈ S.

Since β is a γ-vector space morphism, β(te) = γ(t)ẽ for all t ∈ K, and it thus follows from (43) that α(ta) = α(a · te) = α(a)β(te) = γ(t)α(a) for all t ∈ K and all a ∈ X_{0}, so α is a γ-vector space morphism as well. Also, since ψ is injective, α : X_{0} → X̃_{0} and γ : K → K̃ are injective as well.

It only remains to show that h̃(α(a), α(b)) = β(h(a, b)) for all a, b ∈ X_{0}. By [9, (26.19.i)],

θ(a + b, e) − θ(a, e) − θ(b, e) = h(b, a) − g(a, b)e  (45)

for all a, b ∈ X_{0}, and a similar identity holds in Ω̃. On the other hand, it follows from (42) with t = 0 that θ̃(α(a), ẽ) = β(θ(a, e)) for all a ∈ X_{0}. If we evaluate this identity in a + b with a, b ∈ X_{0}, then it follows from (45) that

h̃(α(b), α(a)) − g̃(α(a), α(b))ẽ = β(h(b, a)) − β(g(a, b)e)

for all a, b ∈ X_{0}. Since β(g(a, b)e) = γ(g(a, b))ẽ = g̃(α(a), α(b))ẽ by (41), we conclude that h̃(α(b), α(a)) = β(h(b, a)) for all a, b ∈ X_{0}, and we are done.

8 Some examples of non-algebraic inclusions of Moufang quadrangles

In this section, we give two examples of inclusions of Moufang quadrangles which are not algebraic.

First assume that Γ_{1} and Γ_{2} are two Moufang quadrangles such that Γ_{1} is a subquadrangle of Γ_{2}, and that none of the root groups is 2-torsion. By Lemma 6.15.(i), the inclusion is either algebraic or dual. We now show that these dual inclusions do really exist.

Theorem 8.1. Let Ω̃ ≅ Ω_{E}(K, V_{0}, q) for some quadratic space (K, V_{0}, q) of type E_{6}, E_{7} or E_{8} with base point e ∈ V_{0}^{∗}, and assume that char(K) ≠ 2. We will write π(a) := θ(a, e) for all a ∈ X_{0}. Let a ∈ X_{0} be arbitrary, let V_{a} be the one-dimensional subspace of V_{0} generated by a, and let q_{a} : V_{a} → K be the quadratic form defined by q_{a}(ta) = t^{2} for all t ∈ K. Let Ω ≅ Ω_{Q}(K, V_{a}, q_{a}). Then Γ(Ω) is isomorphic to a dually included subquadrangle of Γ(Ω̃).

Proof. Let ϕ_{1}, …, ϕ_{4} be the maps defined by

ϕ_{1} : U_{1} → Ũ_{4} : x_{1}(t) ↦ x̃_{4}(te) ,
ϕ_{2} : U_{2} → Ũ_{3} : x_{2}(ta) ↦ x̃_{3}(−ta, 0) ,
ϕ_{3} : U_{3} → Ũ_{2} : x_{3}(t) ↦ x̃_{2}(tπ(a)) ,
ϕ_{4} : U_{4} → Ũ_{1} : x_{4}(ta) ↦ x̃_{1}(ta, 0) ,

for all t ∈ K. Note that x̃_{3}(−ta, 0) = x̃_{3}(ta, 0)^{−1} since char(K) ≠ 2. We have to show that these maps preserve the commutator relations, that is, we have to check whether

ϕ_{2}([x_{1}(t), x_{3}(s)]) = [ϕ_{1}(x_{1}(t)), ϕ_{3}(x_{3}(s))] ,
ϕ_{3}([x_{2}(ta), x_{4}(sa)]) = [ϕ_{2}(x_{2}(ta)), ϕ_{4}(x_{4}(sa))] ,
ϕ_{2}([x_{1}(t), x_{4}(sa)]_{2}) · ϕ_{3}([x_{1}(t), x_{4}(sa)]_{3}) = [ϕ_{1}(x_{1}(t)), ϕ_{4}(x_{4}(sa))] ,

for all s, t ∈ K.

ϕ_{2}([x_{1}(t), x_{3}(s)]) = ϕ_{2}(x_{2}(0)) = x̃_{3}(0, 0) ,
[ϕ_{1}(x_{1}(t)), ϕ_{3}(x_{3}(s))] = [x̃_{4}(te), x̃_{2}(sπ(a))] = [x̃_{2}(sπ(a)), x̃_{4}(−te)^{−1}]^{−1}
= x̃_{3}(0, f(sπ(a), −te))^{−1} = x̃_{3}(0, 0)^{−1} = x̃_{3}(0, 0) ,

for all s, t ∈ K, where we have used the fact that f(e, π(a)) = 0 by [9, (13.41)].

ϕ_{3}([x_{2}(ta), x_{4}(sa)]) = ϕ_{3}([x_{2}(ta), x_{4}(−sa)^{−1}]) = ϕ_{3}(x_{3}(f_{a}(ta, −sa)))
= ϕ_{3}(x_{3}(−2st)) = x̃_{2}(−2stπ(a)) ,
[ϕ_{2}(x_{2}(ta)), ϕ_{4}(x_{4}(sa))] = [x̃_{3}(−ta, 0), x̃_{1}(sa, 0)] = [x̃_{1}(sa, 0), x̃_{3}(ta, 0)^{−1}]^{−1}
= x̃_{2}(h(sa, ta))^{−1} = x̃_{2}(−2stπ(a)) ,

for all s, t ∈ K, where we have used the fact that h(a, a) = 2π(a) by [9, (13.28)].

ϕ_{2}([x_{1}(t), x_{4}(sa)]_{2}) · ϕ_{3}([x_{1}(t), x_{4}(sa)]_{3})

= ϕ_{2}([x_{1}(t), x_{4}(−sa)^{−1}]_{2}) · ϕ_{3}([x_{1}(t), x_{4}(−sa)^{−1}]_{3})

= ϕ_{2}(x_{2}(−tsa)) · ϕ_{3}(x_{3}(tq_{a}(−sa)))
= x̃_{3}(tsa, 0) · x̃_{2}(ts^{2}π(a)) ,
[ϕ_{1}(x_{1}(t)), ϕ_{4}(x_{4}(sa))] = [x̃_{4}(te), x̃_{1}(sa, 0)]
= [x̃_{1}(sa, 0), x̃_{4}(−te)^{−1}]^{−1}
= (x̃_{2}(θ(sa, −te)) · x̃_{3}(sa · (−te), 0))^{−1}
= x̃_{3}(tsa, 0) · x̃_{2}(ts^{2}π(a)) ,

for all s, t ∈ K, where we have used the fact that θ(sa, tv) = s^{2}tθ(a, v) for all a ∈ X_{0}, all v ∈ V_{0} and all s, t ∈ K, by [9, (13.35)]. Hence all the commutator relations are preserved. It is obvious that the maps ϕ_{1}, …, ϕ_{4} are monomorphisms, so we are done.

This is just one example of a dual inclusion; there are definitely more examples, but we do not want to give a classification of all dual inclusions here.

We now give an easy example of a non-algebraic non-dual inclusion; this can only exist if some of the root groups are 2-torsion, so we have to consider algebraic structures where the characteristic of the corresponding (skew) field has characteristic equal to 2.

Theorem 8.2. Let F = GF(2) be the field with 2 elements, and let K := F (α) for some α which is algebraically independent over F. Let V_{0} be a 3-dimensional vector space over K, and let q : V_{0} → K : (y_{1}, y_{2}, y_{3}) ↦ α(y_{1}^{2} + y_{1}y_{2} + y_{2}^{2}) + y_{3}^{2}; then q is an anisotropic quadratic form from V_{0} to K with base point e = (0, 0, 1) ∈ V_{0}^{∗}. Let V_{1} be a 2-dimensional vector space over K, and let p : V_{1} → K : (z_{1}, z_{2}) = (z_{1}^{2} + z_{1}z_{2} + z_{2}^{2}); then q is an anisotropic quadratic form from V_{1} to K with base point d = (1, 0) ∈ V_{1}^{∗}. Let Ω̃ ≅ Ω_{Q}(K, V_{0}, q) and Ω ≅ Ω_{Q}(K, V_{1}, p). Then Γ(Ω) is isomorphic to a non-algebraically and nondually included subquadrangle of Γ(Ω̃).

Proof. Observe that

f_{0}((y_{1}, y_{2}, y_{3}), (z_{1}, z_{2}, z_{3})) = α(y_{1}z_{2} + y_{2}z_{1}) and
f_{1}((y_{1}, y_{2}), (z_{1}, z_{2})) = y_{1}z_{2} + y_{2}z_{1}

for all y_{1}, y_{2}, y_{3}, z_{1}, z_{2}, z_{3} ∈ K. Let ϕ_{1}, …, ϕ_{4} be the maps defined by

ϕ_{1} : U_{1} → Ũ_{1} : x_{1}(t) ↦ x̃_{1}(t) ,
ϕ_{2} : U_{2} → Ũ_{2} : x_{2}(z_{1}, z_{2}) ↦ x̃_{2}(z_{1}, z_{2}, 0) ,
ϕ_{3} : U_{3} → Ũ_{3} : x_{3}(t) ↦ x̃_{3}(αt) ,
ϕ_{4} : U_{4} → Ũ_{4} : x_{4}(z_{1}, z_{2}) ↦ x̃_{4}(z_{1}, z_{2}, 0) ,

for all t, z_{1}, z_{2} ∈ K. Note that all elements of the root groups are equal to their own inverse, since all the root groups are 2-torsion. Again, we start by showing that these maps preserve the commutator relations. The relation [U_{1}, U_{3}] = 1 is obviously preserved. Moreover,

ϕ_{3}([x_{2}(y_{1}, y_{2}), x_{4}(z_{1}, z_{2})]) = ϕ_{3}(x_{3}(y_{1}z_{2} + y_{2}z_{1}))
= x̃_{3}(α(y_{1}z_{2} + y_{2}z_{1})) ,
[ϕ_{2}(x_{2}(y_{1}, y_{2})), ϕ_{4}(x_{4}(z_{1}, z_{2}))] = [x̃_{2}(y_{1}, y_{2}, 0), x̃_{4}(z_{1}, z_{2}, 0)]
= x̃_{3}(α(y_{1}z_{2} + y_{2}z_{1})) ,

for all y_{1}, y_{2}, z_{1}, z_{2} ∈ K, and

ϕ_{2}([x_{1}(t), x_{4}(y_{1}, y_{2})]_{2}) · ϕ_{3}([x_{1}(t), x_{4}(y_{1}, y_{2})]_{3})
= ϕ_{2}(x_{2}(ty_{1}, ty_{2})) · ϕ_{3}(x_{3}(t(y_{1}^{2} + y_{1}y_{2} + y_{2}^{2})))
= x̃_{2}(ty_{1}, ty_{2}, 0) · x̃_{3}(αt(y_{1}^{2} + y_{1}y_{2} + y_{2}^{2})) ,
[ϕ_{1}(x_{1}(t)), ϕ_{4}(x_{4}(y_{1}, y_{2}))] = [x̃_{1}(t), x̃_{4}(y_{1}, y_{2}, 0)]
= x̃_{2}(ty_{1}, ty_{2}, 0) · x̃_{3}(tα(y_{1}^{2} + y_{1}y_{2} + y_{2}^{2})) ,

for all t, y_{1}, y_{2} ∈ K, and hence all commutator relations are preserved.

It remains to show that the inclusion is not algebraic. So suppose that there exists a quadrangular system Ω̃′ = (Ṽ , W̃ , τ_{V} , τ_{W} , ϵ, δ) with corresponding maps F̃ and H̃, and a subsystem Ω′ = (V, W, τ_{V} , τ_{W} , ϵ, δ) of Ω̃′ with corresponding maps F and H, such that Γ(Ω) ≅ Γ(Ω′) and Γ(Ω̃) ≅ Γ(Ω̃′). Since C_{U_{4}}(U_{2}) = 1, Rad(F) = 0, and in particular ϵ ∉ Rad(F). On the other hand, CŨ_{4}(Ũ_{2}) ≠ 1, since it contains the element x_{4}(e), so Rad(F̃) ≠ 0, and by axiom (Q_{10}) in [2, Section 2], ϵ ∈ Rad(F̃) ≤ Rad(F). This contradiction finishes the proof.

The same remark as with the previous example also holds here: There are lots of other examples of non-algebraic non-dual inclusions, but we do not classify them in this paper. In particular, one can construct some more peculiar examples in the exceptional quadrangles of type E_{6}, E_{7} and E_{8}, as well as in those of type F_{4}, but it is out of the scope to go into detail on those examples here.

9 Full and ideal inclusions of Moufang quadrangles

In this final section, we will describe the full and ideal inclusions of two Moufang quadrangles. Definition 9.1. Let Γ_{2} be an arbitrary generalized polygon. Then a subquadrangle Γ_{1} of Γ_{2} is called a full subpolygon if every point row of Γ_{1} coincides with the corresponding point row of Γ_{2}; it is called an ideal subpolygon if every line pencil of Γ_{1} coincides with the corresponding line pencil of Γ_{2}.

Lemma 9.2. Let Γ_{1} and Γ_{2} be Moufang quadrangles for which none of the root groups is 2-torsion, and suppose that Γ_{1} is a full or ideal subquadrangle of Γ_{2}. Then Γ_{1} is algebraically included in Γ_{2}.

Proof. Let Γ_{1} ≅ Γ(Ω) and Γ_{2} ≅ Γ(Ω̃) for some quadrangular systems Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) and Ω̃ = (Ṽ , W̃ , τṼ , τW̃ , ϵ̃, δ̃). By Lemma 6.15.(i), the inclusion of Γ_{1} in Γ_{2} is either algebraic or dual.

So suppose that it is dual, and let ϕ_{i} : U_{i} ↪ Ũ_{5−i} for i ∈ {1, …, 4} be group monomorphisms preserving the commutator relations. In particular,

[ϕ_{1}(U_{1}), ϕ_{3}(U_{3})] = 1 and  (46)
[ϕ_{2}(U_{2}), ϕ_{4}(U_{4})] ≠ 1 .  (47)

It follows from (47) that [Ũ_{1}, Ũ_{3}] ≠ 1, hence Ω̃ is wide, so it is either of pseudoquadratic form type, or of type E_{6}, E_{7} or E_{8}; in particular, W̃ is not abelian. Since V is abelian, we cannot have that V ≅ W̃, and in particular, ϕ_{2}(U_{2}) ≠ Ũ_{3} and ϕ_{4}(U_{4}) ≠ Ũ_{1}. But since the inclusion is full or ideal, we must then have ϕ_{1}(U_{1}) = Ũ_{4} and ϕ_{3}(U_{3}) = Ũ_{2}. It then follows from (46) that [Ũ_{2}, Ũ_{4}] = 1, which contradicts the fact that none of the root groups is 2-torsion. Hence the inclusion cannot be dual, so it must be algebraic.

Definition 9.3. Let Ω̃ = (Ṽ , W̃ , τ_{V} , τ_{W} , ϵ, δ) be a quadrangular system, and let Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) be a subsystem of Ω̃ such that the corresponding inclusion of Moufang quadrangles is full or ideal. Then we say that Ω is a full subsystem of Ω̃. If V = Ṽ, then we call Ω a V -full subsystem of Ω̃, and if W = W̃, then we call it a W -full subsystem of Ω̃.

We now consider an arbitrary quadrangular system of a certain type, and we examine which quadrangular systems can occur as full subsystems.

Theorem 9.4. (i) Let Ω̃ ≅ Ω_{I}(K, σ) for some proper involutory set (K, σ). Then Ω̃ does not have full subsystems other than Ω̃ itself.

(ii) Let Ω̃ ≅ Ω_{Q}(K, V_{0}, q) for some anisotropic quadratic space (K, V_{0}, q). Then every W -full subsystem of Ω̃ is of the form Ω ≅ Ω_{Q}(K, V_{1}, q|_{V_{1}}) for some subspace V_{1} of V_{0}; Ω̃ does not have V -full subsystems other than Ω̃ itself.

(iii) Let Ω̃ ≅ Ω_{P} (K, σ, V_{0}, π) for some proper pseudo-quadratic space (K, σ, V_{0}, π). Then every V -full subsystem of Ω̃ is of the form Ω ≅ Ω_{P} (K, σ, V_{1}, π|_{V_{1}}) for some subspace V_{1} of V_{0} (where Ω might or might not be proper); Ω̃ does not have W -full subsystems other than Ω̃ itself.

(iv) Let Ω̃ ≅ Ω_{E}(K, V_{0}, q) for some quadratic space (K, V_{0}, q) of type E_{6}, E_{7} or E_{8}. Then there is only one proper V -full subsystem of Ω̃, namely Ω ≅ Ω_{Q}(K, V_{0}, q); Ω̃ does not have W -full subsystems other than Ω̃ itself.

Proof. Let Ω̃ = (Ṽ , W̃ , τ_{V} , τ_{W} , ϵ, δ) be a quadrangular system, and let Ω = (V, W, τ_{V} , τ_{W} , ϵ, δ) be a subsystem of Ω̃.

(i) Suppose that Ω̃ ≅ Ω_{I}(K, σ). Then F̃(Ṽ , Ṽ) = W̃ since char(K) ≠ 2. In particular, if V = Ṽ, then it follows from F (V, V) ≤ W that W = W̃ as well, hence Ω = Ω̃.

On the other hand, suppose that W = W̃. Since (K, σ) is proper, K is generated by Fix_{K}(σ), and hence every element v ∈ Ṽ = [K] can be written as v = ϵw_{1} ⋯ w_{n} for some n and some elements w_{1}, …, w_{n} ∈ W̃ = [Fix_{K}(σ)]. But this implies that v ∈ ϵW ⋯ W ≤ V, so V = Ṽ as well, and again, we conclude that Ω = Ω̃.

(ii) Suppose that Ω̃ ≅ Ω_{Q}(K, V_{0}, q) for some anisotropic quadratic space (K, V_{0}, q). By Theorem 7.10, every subsystem of Ω̃ has to be of quadratic form type as well. Since char(K) ≠ 2, the bilinear form f corresponding to q is not identically zero, and hence surjective, so F̃(Ṽ , Ṽ) = W̃. Again, it follows that every V -full subsystem is also W -full and hence equal to Ω̃ itself.

So suppose that W = W̃. It then follows that Ω ≅ Ω_{Q}(K, V_{1}, p) for some subspace V_{1} ≤ V_{0}, and since τ_{W} ([t], [v]) = [tq(v)] for all t ∈ K and all v ∈ V_{0}, and similarly τ_{W} ([t], [v]) = [tp(v)] for all t ∈ K and all v ∈ V_{1}, it follows that p is the restriction of q to V_{1}, which is what we had to show.

Note that, in principle, we have to require V_{1} to contain the base point of (K, V_{0}, q), but since the base point can be chosen arbitrarily (see Remark 7.7), this restriction is obsolete.

(iii) Suppose that Ω̃ ≅ Ω_{P} (K, σ, V_{0}, π) for some proper pseudo-quadratic space (K, σ, V_{0}, π). Suppose that W = W̃. Since (K, σ, V_{0}, π) is proper, the corresponding skew-hermitian map h : V_{0}×V_{0} → K is not identically zero, and hence onto. Therefore H̃(W̃ , W̃) = Ṽ, and it follows that V = Ṽ as well, so Ω = Ω̃.

So assume that V = Ṽ. Suppose first that Ω is of involutory type. Since V = Ṽ = [K], we must have Ω ≅ Ω_{I}(K, σ′) for some involution σ′ of K.

But if we evaluate ϵF (ϵ, [t]) − [t] in Ω and in Ω̃, then we get that σ′ = σ, and hence Ω ≅ Ω_{I}(K, σ) ≅ Ω_{P} (K, σ, 0, 0).

Next, suppose that Ω is of quadratic form type. By Theorem 7.20, Ω is a subsystem of Ω_{I}(K, σ), and by Theorem 7.18, the involutory set (K, σ) is not proper, and hence Ω_{I}(K, σ) is of quadratic form type. Since Ω is a V -full subsystem of Ω_{I}(K, σ), it follows from (ii) of this theorem that we must have Ω ≅ Ω_{I}(K, σ) ≅ Ω_{P} (K, σ, 0, 0).

Now suppose that Ω is of proper pseudo-quadratic form type. Since V = Ṽ = [K], we must have Ω ≅ Ω_{P} (K, σ′, V_{1}, π′) for some involution σ′ of K, some vector space V_{1} over K and some anisotropic pseudo-quadratic form π′ from V_{1} to K. Since Ω_{I}(K, σ′) is then also a V -full subsystem of Ω̃, it follows as before that σ′ = σ. It follows from the relation [a, 0][t] = [at, 0] for all a ∈ V_{0} and all v ∈ K that the additive subgroup V_{1} ≤ V_{0} is in fact a K-subspace of V_{0}. Since π′(v) = h(v, v)/2 = π(v) for all v ∈ V_{1}, π′ is the restriction of π to V_{1}, and hence we have shown that Ω is as required.

Finally, Ω cannot be of type E_{6}, E_{7} or E_{8} by Theorem 7.22.

(iv) Suppose that Ω̃ ≅ Ω_{E}(K, V_{0}, q) for some quadratic space of type E_{6}, E_{7} or E_{8}. Suppose that W = W̃. By [9, (13.25)], the corresponding map h : X_{0} × X_{0} → V_{0} is surjective. Therefore H̃(W̃ , W̃) = Ṽ, and it follows that V = Ṽ as well, so Ω = Ω̃.

So assume that V = Ṽ. We have that F̃(Ṽ , Ṽ) = [0, K] < [S] = W̃, and hence F (V, V) = [0, K] as well; in particular, [0, K] ≤ W. If W = [0, K], then Ω ≅ Ω_{Q}(K, V_{0}, q). So we may now assume that there exists an element (ξ, t) ∈ S such that [ξ, t] ∈ W. Since [0, t] ∈ W as well, we also have that [ξ, 0] ∈ W. It then follows from the relation [a, 0][v] = [av, 0] for all a ∈ X_{0} and all v ∈ V_{0} that [ξv, 0] ∈ W. Continuing in this way, we obtain that [ξV_{0} ⋯ V_{0}, 0] ≤ W. By [9, (27.7)], however, ξC_{0}(V_{0}, q) = X_{0}, and hence [X_{0}, 0] ≤ W. Again using the fact that [0, K] ≤ W, we conclude that W = W̃, and hence Ω = Ω̃.

Acknowledgments

I thank Hendrik Van Maldeghem and Richard Weiss for making some very valuable comments on this matter. I am also grateful to Jean-Pierre Tignol for communicating the example in Remark 7.19 to me.

References

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[2] T. De Medts: An algebraic structure for Moufang quadrangles, submitted.

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[8] J.-P. Tignol: Réduction de l’indice d’une algèbre simple centrale sur le corps des fonctions d’une quadrique, Bull. Soc. Math. Belg., Sér. A 42, 1990, pp. 735–745.

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[11] M. Wolfrom: “Einbettungen zwischen klassischen Vierecken”, Diplomarbeit, Mathematisches Institut der Bayerischen Julius-Maximilians-Universität Würzburg, 1998.

Tom De Medts Department of Pure Mathematics and Computer Algebra Ghent University Galglaan 2, B-9000 Gent Belgium

e-mail: tdemedts@cage.UGent.be