A new construction of Moufang quadrangles of type E6, E7 and E8

In the classification of Moufang polygons by J. Tits and R. Weiss, the most intricate case is by far the case of the exceptional Moufang quadrangles of type E_6, E_7 and E_8, and in fact, the construction that they present is ad-hoc and lacking a deeper explanation. We will show how tensor products of two composition algebras can be used to construct these Moufang quadrangles in characteristic different from 2. As a byproduct, we will obtain a method to construct any Moufang quadrangle in characteristic different from two from a module for a Jordan algebra.


Introduction
In the late sixties, Jacques Tits introduced an (at that time) innovative tool to study semisimple linear algebraic groups of positive relative rank, namely the theory of spherical buildings. Especially in the case of the exceptional groups, these buildings are quite often a main effective tool, and the algebraic description that goes with them, is invaluable in order to perform explicit calculations involving exceptional groups.
The cases where the relative rank is at least two are relatively well understood, mainly because of the work of Jacques Tits and Richard Weiss in the theory of Moufang polygons [TW]. However, for the Moufang quadrangles of exceptional type, mainly those of type E 6 , E 7 and E 8 , the construction is rather ad-hoc, and a deeper explanation is still missing.
The goal of our paper is to give an explicit but at the same time completely intrinsic method to construct a family of rank two groups corresponding to the Moufang quadrangles of type E 6 , E 7 and E 8 in characteristic different from two.
We will be dealing with the forms given by the following Tits indices: Observe that all three forms have the property that their anisotropic kernel is of type D n (with an additional factor A 1 for the form of type E 7 ); this implies that these forms will be determined by an anisotropic quadratic form of dimension 2n having certain additional properties. (The A 1 factor in the E 7 case gives rise to a quaternion division algebra, which turns up in the description of the Hasse invariant of the quadratic form corresponding to the D n factor.) We will refer to such quadratic forms as forms of type E 6 , E 7 and E 8 , respectively.
In each case, we will see that the quadratic form can be characterized as the anisotropic part of the Albert form of a certain tensor product of composition algebras, and in fact, these algebras themselves will play a crucial role in the understanding of the corresponding algebraic groups; they completely determine the algebraic group up to isogeny.
Our approach will turn out to be applicable in a more general situation, and in fact, we will obtain every possible Moufang quadrangle defined over a field of characteristic different from two starting from certain modules over a Jordan algebra. Our construction relies in an essential way on the theory of J-ternary algebras and their Peirce decomposition, see [ABG,Sections 3.12 and 6.61].
We can summarize our main result, namely the explicit construction of the quadrangular algebras of type E 6 , E 7 and E 8 , as follows.
Construction. Let char(k) = 2. We start with a quadratic space (k, V, q) of type E 6 , E 7 or E 8 with base point (see also Definition 2.15 below). By Theorem 3.11, there exist an octonion division algebra C 1 and a division composition algebra C 2 of dimension 2, 4 or 8, respectively such that C 1 and C 2 contain an isomorphic quadratic field extension, but no isomorphic quaternion algebra, and such that q is similar to the anisotropic part of the Albert form, q A , of C 1 ⊗ k C 2 . It follows that there exist i 1 ∈ C 1 and i 2 ∈ C 2 such that i 2 1 = i 2 2 = a ∈ k \ {k 2 }. We define a subspace V of the skew-elements of C 1 ⊗ k C 2 of dimension 6, 8 or 12, respectively, as 1 We choose an arbitrary u ∈ V \ {0} and define the quadratic form this form has base point u and is similar to the quadratic form of type E 6 , E 7 or E 8 we started with.
We then define the subspace X 0 of C 1 ⊗ k C 2 of dimension 8, 16 or 32 as X 0 := x ⊗ y + 1 a i 1 x ⊗ i 2 y | x ∈ C 1 , y ∈ C 2 .
Next, we define a suitable element r ∈ S as in Definition 3.16(iii) below, and we define the bilinear map X 0 × L 0 → X 0 as x · v = v(r(u(rx))), and the bilinear map h : X 0 × X 0 → V as h(x, y) = (u(rx))y − y((xr)u).
In Theorem 3.20 we prove that the 7-tuple (k, V, Q, u, X 0 , ·, h) is a quadrangular algebra of type E 6 , E 7 or E 8 , respectively. It follows that this is the structure described in [TW,Chapter 13] giving rise to the Moufang quadrangles of type E 6 , E 7 and E 8 , and hence to the corresponding rank two forms of exceptional linear algebraic groups of type E 6 , E 7 and E 8 .
Organization of the paper In Section 2 we give some preliminar material on composition algebras, tensor products of composition algebras, quadrangular algebras and Peirce decomposition in Jordan algebras.
In Section 3 we work towards the main theorem of our paper, which is Theorem 3.20 and which gives a construction of quadrangular algebras of type E 6 , E 7 and E 8 .
In Section 3.1 we show that we can construct a quadrangular algebra in characteristic not 2 starting from a specific kind of module for a Jordan algebra of reduced spin type (see Definition 2.20).
In Section 3.2 we construct, in a similar way, a quadrangular algebra starting from a J-ternary algebra over a Jordan algebra of reduced spin type. This section deals only with fields of characteristic not 2 nor 3, since J-ternary algebras are only defined over such fields.
In Section 3.3 we show that we can apply this procedure to obtain each quadrangular algebra of pseudo-quadratic form type in characteristic not 2. In Section 3.4 we construct (see the construction above) a module for a Jordan algebra of reduced spin type out of the tensor product of two composition algebras and show that this gives rise to quadrangular algebras of type E 6 , E 7 and E 8 in characteristic not 2.
In Section 4, inspired by Theorem 3.5, we give a uniform description of all Moufang quadrangles in characteristic different from two.
Acknowledgments Some of our ideas were inspired by fruitful discussions with Skip Garibaldi, in particular during a longer visit of the first author at Emory University, whose hospitality is gratefully acknowledged. Skip Garibaldi's observation mentioned in Theorem 3.11 was a crucial first step in the whole project. We also thank Bruce Allison for fruitful discussions about J-ternary algebras. Last but not least, we are greatly indebted to the referee for a spectacularly detailed and careful reading of our paper.

Preliminaries
We assume throughout the paper that k is a commutative field of characteristic different from 2.

Composition algebras
A composition algebra is a, not necessarily commutative nor associative, unital k-algebra C equipped with a quadratic form q : C → k that is multiplicative, i.e. q(xy) = q(x)q(y) for all x, y ∈ C. This quadratic form q is called the norm form, its associated bilinear form will be denoted by f . With the norm form we associate an involution on C by defining By a classical result (see for example [SV,Theorem 1.6.2]) each composition algebra has dimension 1, 2, 4 or 8: (i) If dim k C = 1, then C = k, q(x) = x 2 and the involution is trivial. (ii) If dim k C = 2, then C/k is a quadraticétale extension of k. There exists a ∈ k such that C = k[i]/(i 2 − a), the norm form is 1, −a . Either C/k is a separable quadratic field extension and σ is the nontrivial element of Gal(C/k), or C ∼ = k ⊕ k and σ interchanges the two components.
This quaternion algebra is denoted by (a, b) k . The norm form is equal to 1, −a 1, −b , the involution fixes k and maps i → −i, j → −j. (iv) If dim k C = 8, then C/k is an octonion algebra over k. There exist a, b, c ∈ k such that C = Q ⊕ Qk where Q = (a, b) k and multiplication is given by (x 1 +x 2 k)(y 1 +y 2 k) = (x 1 y 1 +cy 2 x 2 )+(y 2 x 1 +x 2 y 1 )k for all x i , y i ∈ Q.
The norm form is 1, −a 1, −b 1, −c and the involution is given by In each case, the norm form is a Pfister form, these are forms of dimension 2 n denoted by a 1 , . . . , a n := ⊗ n i=1 1, a i for a 1 , . . . , a n ∈ k. The norm form is anisotropic when C is a division algebra, and it is hyperbolic otherwise (i.e. when C is a split algebra).
The norm form is completely determined by the algebra structure of the composition algebra. It is a well known but somewhat deeper fact (see e.g. [SV]) that the converse also holds, i.e. the composition algebra is determined up to isomorphism by the (similarity class of) the norm.
Quaternion algebras are not commutative, but associative. Octonion algebras are neither commutative nor associative. In the lemma below we summarize some useful identities that hold in each composition algebra.
Property (iii) is called the alternativity, the identities in (v) are called the Moufang identities.

Tensor products of composition algebras
We now assume that C 1 and C 2 are two composition algebras over k (possibly of different dimension), with norm forms q 1 and q 2 and involutions σ 1 and σ 2 , respectively. Consider If char(k) = 2, 3 the algebra (C 1 ⊗ k C 2 , σ 1 ⊗ σ 2 ) is a structurable algebra. This is a class of algebras that generalizes Jordan algebras and associative algebras with involution. We will not need the exact definition and refer the interested reader to [A1].
Let S i be the set of skew elements in C i , i.e.
and similarly, let S be the set of skew elements of C 1 ⊗ k C 2 , i.e.
Definition 2.2. We will associate a quadratic form q A to C 1 ⊗ k C 2 , called the Albert form, by setting for all x ∈ S 1 and y ∈ S 2 . When we denote q ′ i = q i | S i for the pure part of the Pfister form q i , we have This form is named after A.A. Albert, who studied the case where C 1 and C 2 are both quaternion algebras, i.e. C 1 ⊗ k C 2 is a biquaternion algebra.
Definition 2.3. Let s = s 1 ⊗ 1 + 1 ⊗ s 2 ∈ S, we define the map sharp by If q A (s) = 0, the inverse of s is defined by Tensor products of two composition algebras are far from associative or alternative, but the skew elements behave nicer than arbitrary elements: Proof. These identities can be easily checked using Lemma 2.1.
Remark 2.5. In the case that both C 1 and C 2 are quaternion algebras, C 1 ⊗ k C 2 is associative and A.A. Albert proved that C 1 ⊗ k C 2 is a division algebra if and only if its Albert form is anisotropic (see [L,Theorem III.4.8].) It is not obvious to generalize this result to arbitrary composition algebras. (Notice that, in the theory of structurable algebras, the concept of conjugate invertibility is used.) In [A2,Theorem 5.1] it is proven in the case that char(k) = 0 that the tensor product of two octonion algebras is a conjugate division algebra if and only if the corresponding Albert form is anisotropic. To the best of our knowledge, it is an open problem whether this equivalence also holds for fields of characteristic > 3.
The case where q A has Witt index one will be needed to study the rank two forms of linear algebraic groups of type E 6 , E 7 , E 8 discussed in the introduction.
Definition 2.6 ( [L,Definition 5.11]). Two n-fold Pfister forms q 1 , q 2 are rlinked if there is an r-fold Pfister form h such that q 1 ≃ h⊗q 3 and q 2 ≃ h⊗q 4 for some Pfister forms q 3 , q 4 .
The linkage number of q 1 and q 2 is the number r ∈ N such that q 1 and q 2 are r-linked but not (r + 1)-linked.
Lemma 2.7. Let C 1 be an octonion division algebra with norm q 1 and let C 2 be a separable quadratic field extension, quaternion division algebra or an octonion division algebra, with norm q 2 . The following are equivalent: (i) C 1 and C 2 contain isomorphic separable quadratic field extensions, but C 1 and C 2 do not contain isomorphic quaternion algebras. (ii) The linkage number of q 1 and q 2 is 1, i.e. q 1 and q 2 are 1-linked but not 2-linked. (iii) The Witt index of the Albert form q A of C 1 ⊗ k C 2 is equal to one.
Proof. Since the Witt index of q A is one less than the Witt index of q 1 ⊥ −q 2 , the equivalence of (ii) and (iii) is given by a result of Elman-Lam (see for example [L,Theorem X.5.13]).
The following observations follow from [SV,Prop. 1.5.1]. Let C be a composition algebra over k with norm q.
Let dim(C) = 4 or 8. Then C contains a separable extension field isomorphic to k(i)/(i 2 − a) with a ∈ k if and only if there exists a Pfister form ϕ, of dimension 2 or 4 respectively, such that q ≃ −a ⊗ ϕ.
Let dim(C) = 8. Then C contains a quaternion algebra isomorphic to (a, b) k with a, b ∈ k if and only if q ≃ −a, −b, −c for some c ∈ k.
From this it follows immediately that (i) and (ii) are equivalent.
Remark 2.8. Suppose that C 1 and C 2 contain isomorphic separable quadratic field extensions. Even if they do not contain isomorphic quaternion algebras, it is still possible that C 1 and C 2 contain more than one isomorphic separable quadratic field extension up to isomorphism.
Definition 2.9. We define the linkage number of C 1 and C 2 as the linkage number of their norm forms q 1 and q 2 .
Lemma 2.7 indicates that we will be particularly interested in pairs of composition algebras C 1 , C 2 with linkage number one.

Quadrangular algebras
A quadrangular algebra is an algebraic structure that was constructed to describe the exceptional Moufang quadrangles. In Section 4 we explain how one constructs Moufang quadrangles out of quadrangular algebras. For more information on quadrangular algebras, including characteristic 2, we refer to [W]. We emphasize that the structure of a quadrangular algebra simplifies significantly in characteristic different from 2, see Remark 2.11. Since this is the only case we will be dealing with we restrict our definition to char(k) = 2.
Definition 2.10. A quadrangular algebra, in characteristic different from 2, is an 7-tuple (k, L, q, 1, X, ·, h), where (i) k is a commutative field with char(k) = 2, (ii) L is a k-vector space, (iii) q is an anisotropic quadratic form from L to k, (iv) 1 ∈ L is a base point for q, i.e. an element such that q(1) = 1, (v) X is a non-trivial k-vector space, (vi) (x, v) → x · v is a map from X × L to X (usually denoted simply by juxtaposition), (vii) h is a map from X × X to L, satisfying the following axioms, where for all x, y ∈ X and all v ∈ L.
Moreover, we define a map g : X × X → k by for all x, y ∈ X.
Remark 2.11. When one compares our definition of quadrangular algebras with the general definition in [W,Definition 1.17] there are two differences which are due to the fact that the definition simplifies when the characteristic is different from 2.
Theorem 2.12. A quadrangular algebra in characteristic not 2 is either obtained from an anisotropic pseudo-quadratic space over a quadratic pair (see Section 2.3.1) or is of type E 6 , E 7 or E 8 (see Section 2.3.2).
Proof. Since the characteristic of k is not 2, it follows from [W, 2.3 and 2.4] that the quadrangular algebra is regular, i.e. f is non-degenerate (from [W, 3.14] it follows that it is also proper, i.e. σ = 1). Now it follows from [W, 3.2] that if the quadrangular algebra is not special (i.e. not arising from a pseudo-quadratic space) it is of type E 6 , E 7 or E 8 .

Pseudo-quadratic spaces
Definition 2.13 ([W, Definition 1.16]). A pseudo-quadratic space over a field of characteristic not 2 is a quintuple (L, σ, X, h, π) where (i) L is a skew field of characteristic different from 2; (ii) σ is an involution of L, and we let • h is bi-additive and h(x, yu) = h(x, y)u, and • h(x, y) σ = −h(y, x), for all x, y ∈ X and all u ∈ L; (v) π is a pseudo-quadratic form from X to L, i.e.
• π(x + y) ≡ π(x) + π(y) + h(x, y) mod L σ , and • π(xu) ≡ u σ π(x)u mod L σ , for all x, y ∈ X and all u ∈ L. Since we work in characteristic not 2 we can always assume that the pseudo-quadratic space is standard, i.e. π(x) = 1 2 h(x, x). A pseudo-quadratic space (L, σ, X, h, π) is called anisotropic if Not every pseudo-quadratic space is a quadrangular algebra; to be a pseudo-quadratic space the skew field has to satisfy some additional properties.
Definition 2.14 ([W, Definition 1.12]). Let L be a skew-field with involution σ. We call (L, σ) a quadratic pair 2 , if k := L σ is a field and if either (i) L/k is a separable quadratic field extension and σ is the generator of the Galois group; or (ii) L is a quaternion algebra over k and σ is the standard involution.
A result of Dieudonné (see for example [W,Theorem 1.15]) says that if σ is not trivial, the either L is generated by L σ as a ring or (L, σ) is a quadratic pair (in this case L σ is a field). From this point of view quadratic pairs are an exceptional class of skew-fields.
In [W,Proposition 1.18] it is shown that a non-zero standard anisotropic pseudo-quadratic space over a quadratic pair gives rise to the quadrangular algebra (k, L, q, 1, X, scalar multiplication, h).
2.3.2 Quadrangular algebras of type E 6 , E 7 and E 8 For an explicit description of quadrangular algebras of type E 6 , E 7 and E 8 , we refer to [TW,Chapter 12 and 13]; for a concise description we refer to the first part of [W,Chapter 10]. Some care is needed, since the map g in [TW] is equal to −g in [W]. Here we only give a concise overview of the structure of a quadrangular algebra of type E 6 , E 7 or E 8 .
We always assume that s 2 s 3 s 4 s 5 s 6 = −1, which can be achieved by rescaling the quadratic form if necessary.
As we are working in characteristic not 2, we can choose γ ∈ E such that E = k(γ) and γ 2 ∈ k.
If (k, L, q) is a quadratic space of type E 6 , E 7 or E 8 with base point, there exists a scalar multiplication E × L → L that extends the scalar multiplication k × L → L.
Let (k, L, q, 1, X, ·, h) be a quadrangular algebra of type E 6 , E 7 or E 8 , then (k, L, q) is a quadratic space of type E 6 , E 7 or E 8 , respectively with basepoint denoted by 1. This quadratic space determines the quadrangular algebra entirely (see [W,Theorem 6.24]).
The vector space X has k-dimension 8, 16 or 32, respectively; it is a C(q, 1)-module (see Definition 2.16 below). Some of the properties of the maps ·, h, θ and π are given in Definition 2.10. The existence of the vector space X and of the maps ·, h and θ is shown in [TW,Chapter 13] by giving an explicit ad-hoc construction using the coordinatization of L.
The goal of this article is to provide an alternative description of X, L and the maps ·, h starting from the tensor product of composition algebras.
In order to prove the anisotropy of our new construction of the map π (see Theorem 3.20), we need the concept of an irreducible C(q, 1)-module.
Definition 2.16. (i) Let (k, V, q) be a quadratic space with basepoint 1, then the Clifford algebra of q with basepoint 1 is defined as where T (V ) is the tensor algebra of V , and where σ is defined as in Definition 2.10. It is shown in [TW,12.51] that C(q, 1) ∼ = C 0 (q), the even Clifford algebra of q. The notion of a Clifford algebra with base point was introduced by Jacobson and McCrimmon; see [TW,Chapter 12] for more details. (ii) Since q is anisotropic, the axioms (A1)-(A3) of an arbitrary quadrangular algebra say precisely that X is a C(q, 1)-module, such that the action of C(q, 1) on X is an extension of the action of L on X (see [W,Proposition 2.22]).
Theorem 2.17 ( [W, 2.26]). Let (k, V, q) be a quadratic space of type E 6 , E 7 or E 8 with basepoint 1 and let X be a right C(q, 1)-module. Then X is an irreducible right C(q, 1)-module if and only if dim k (X) = 8, 16 or 32, respectively.

Peirce decomposition in Jordan algebras
A good reference to study the theory of Jordan algebras is [M]. Our construction of exceptional quadrangular algebras uses the Peirce decomposition of a Jordan algebra. We summarize the main properties and multiplication rules of Peirce subspaces.
Definition 2.18. A Jordan k-algebra J is a unital commutative k-algebra such that for all x, y ∈ J we have (x 2 y)x = x 2 (yx).
We define the U -operator and its linearization for x, y, z ∈ J An element x ∈ J is invertible if and only if there exists a y ∈ J such that xy = 1 and x 2 y = x; this condition is equivalent with U x y = x, U x y 2 = 1. The element y is the inverse of x.
For a nondegenerate Jordan algebra we have J 0 = 0 (see [M,II.10.1.2]). Let i ∈ {0, 1} and j = 1−i; then To construct quadrangular algebras we will use two types of Jordan algebras that contain supplementary idempotents. These are the Jordan algebras of reduced spin type and the Jordan algebras H(M 2 (L), σt) for a skew field L with involution σ, where t is the transpose map.
Definition 2.20 ([M, II.3.4 on p. 180]). Consider a quadratic form q : V → k over k. Starting from the vector space V we construct a Jordan algebra by adjoining two supplementary idempotents to V .
As a vector space, we define J by adjoining two copies of k to V : J := ke 0 ⊕ V ⊕ ke 1 . We define the following multiplication: This defines a Jordan algebra 3 on ke 0 ⊕ V ⊕ ke 1 , called the reduced spin factor of the quadratic form q. We say that J is of reduced spin type.
The unit of this Jordan algebra is e 0 + e 1 , and for all v, w ∈ V we have It is clear that e 0 and e 1 are supplementary idempotents and that we have the following Peirce subspaces with respect to e 1 : We define the supplementary idempotents We have Remark 2.22. If we consider the above definition in the case that (L, σ) is a quadratic pair (see Definition 2.14), then k = L σ . Now there exists a non-degenerate anisotropic quadratic form q : L → k : ℓ → ℓℓ σ = ℓ σ ℓ and the Peirce subspaces . By comparing the multiplication in Definition 2.21 above and the one in Definition 2.20 we conclude that H(M 2 (L), σt) is the reduced spin factor of the quadratic space (J 1 / 2 , k, Q).
In the following Proposition we use Osborn's Capacity Two theorem to show that the two families of Jordan algebras we discussed above can be characterized in a unified way. The proof of this Proposition uses some results and concepts of Jordan theory that we will not use in the remaining part of this article.
Proposition 2.23. Let J be a non-degenerate Jordan k-algebra with supplementary proper idempotents e 0 and e 1 . Let J 0 , J 1 / 2 , J 1 be the Peirce subspaces of J with respect to e 1 . We assume that each element in J 1 / 2 \ {0} is invertible and that there exists u ∈ J 1 / 2 such that u 2 = 1.
• If dim(J 0 ) = 1, then J is the reduced spin factor of some non-degenerate anisotropic quadratic space with basepoint u. 5 • If dim(J 0 ) > 1, then J is isomorphic to H(M 2 (L), σt) for some skew field L with involution σ, such that (L, σ) is not a quadratic pair.
Proof. We will show that the assumptions imply that J is a simple nondegenerate Jordan algebra of capacity 2; an algebra has capacity 2 if the unit is the sum of two supplementary idempotents e 0 , e 1 such that the Peirce subspaces J 0 , J 1 are division algebras.
Since u 2 = 1 it follows from [M, II.6.1.10] that U u is a Jordan isomorphism 6 of J such that (U u ) 2 is the identity map. Since U u (J 1 ) ⊆ J 0 , U u is an isomorphism between J 0 and J 1 . Therefore it is enough to show that J 0 is a division algebra. It follows from [M,II.6 and hence U t is surjective on J 0 .
This proves that J has capacity 2.
We proved all the conditions of Osborn's Capacity Two theorem, see [M,II.22.2.1 on p. 351]. This theorem states that a simple nondegenerate Jordan algebra of capacity 2 belongs to exactly one of the following three disjoint classes from which we can exclude the first: then it would follow that vw = 0 for all w ∈ J 1 / 2 , which implies that v is not invertible. Therefore q is anisotropic. In this case dim(J 0 ) = 1.

A coordinate-free construction of quadrangular algebras
In this section we give a coordinate-free construction of quadrangular algebras. Our construction was inspired by several properties of J-ternary algebras; see Definition 3.6 and [ABG,3.12].
The entire article [ABG] deals with fields of characteristic zero only. However the concept of a J-ternary algebra and its basic properties, such as Peirce decomposition (see Lemma 3.4 and [ABG, 6.61]) can be generalized without any adjustments to fields of characteristic different from 2 and 3.
It is not clear at all how to generalize the theory of J-ternary algebras to fields of characteristic 2 and 3; one reason is that the definition of a J-ternary algebra uses bilinear and trilinear forms.
However in the theory of quadrangular algebras there is no difference between fields of characteristic different form 2 and 3 and fields of characteristic 3. Therefore we want our construction of quadrangular algebras to work in characteristic 3 in the same way as in characteristic not 2 and 3.
Actually, J-ternary algebras contain some axioms that are superfluous for our construction. In Theorem 3.5 we show that we can prove the axioms for quadrangular algebras using only a few axioms concerning a module for a Jordan algebra that has a skew-symmetric form. We replaced all the identities in the definition of a J-ternary system involving the trilinear form by an identity that only uses the bilinear form. Therefore we do not need the assumption that the characteristic is different from 3. In Section 3.2 we show that a J-ternary algebra still satisfies the identity we demand in Theorem 3.5.
In the following subsection we do not start by giving a definition of Jternary algebras. Instead we start by considering the concepts that we will need to formulate Theorem 3.5.
To include characteristic 2 would be another cup of tea for several reasons: already the definition of quadrangular algebras is much more complicated and the definition of a Jordan algebra is more delicate.

Quadrangular algebras from special Jordan modules
Let k be a field of characteristic different from 2. In the next lemma we introduce a module for Jordan algebras.
Lemma 3.1. Let J be a Jordan k-algebra and let X be a k-vector space. Suppose that J acts on X by • : J × X → X such that for all j, j ′ ∈ J, x, y ∈ X, t ∈ k. Then the following are equivalent Linearizing this expression gives us (v').
Assume that (v') holds, we have Definition 3.2 ( [ABG,3.12]). Let J be a Jordan k-algebra and let X be a k-vector space with action • : J × X → X. Then X is a special J-module if the conditions (i)-(v) of the previous lemma are satisfied.
Lemma 3.3. Let X be a special J-module and let ( ( (·, ·) ) ) : X × X → J be a bilinear skew-symmetric form. Then In the following Lemma we consider the Peirce decomposition of special J-modules; see also [ABG,6.61].
Lemma 3.4. Let J be a Jordan k-algebra with supplementary proper idempotents e 0 and e 1 . Let J 0 , J 1 / 2 , J 1 be the Peirce subspaces of J with respect to e 1 . Let X be a special J-module and define for i ∈ {0, 1} and j = 1 − i. (iv) Assume there exists an element u ∈ J 1 / 2 such that u 2 = 1. The map is a vector space isomorphism, called the connecting morphism.
x the connecting morphism is an isomorphism.
Let J be the reduced spin factor of the non-degenerate anisotropic quadratic space (k, V, q) with base point u: J = ke 0 ⊕ V ⊕ ke 1 .
Assume that the following holds: Then (k, V, q, u, X 0 , ·, h) is a quadrangular algebra.
Proof. Notice that e 0 , e 1 ∈ J are supplementary proper idempotents and that u ∈ J 1 / 2 such that u 2 = q(u)1 = 1. Thus we can apply Lemma 3.4 with J 0 = ke 0 , J 1 / 2 = V, J 1 = ke 1 . It follows from (3.2) and (3.3) that the maps · and h are well defined. To start we show that U u (v) = v σ for all v ∈ J 1 / 2 with σ as in Definition 2.10: We verify that all the axioms of a quadrangular algebra given in Definition 2.10 hold.
Since this expression is linear in v and trivial for v ∈ ku, we can assume v ∈ u ⊥ and thus f (u, v) = uv = 0 and hence u In this case we continue as follows: This is exactly (3.4). (D2) This is assumption (3.5).

Quadrangular algebras from J-ternary algebras
In this subsection we assume char(k) = 2, 3 and we prove that an arbitrary 'anisotropic' non-trivial J-ternary algebra, where J is as in Theorem 3.5, satisfies the assumptions of Theorem 3.5. It follows that we can construct quadrangular algebras from J-ternary algebras.
We remind the reader that the entire article [ABG] is only written for fields of characteristic zero, but that the concept of a J-ternary algebra and its basic properties can be generalized without any adjustments to hold in fields of characteristic different from 2 and 3.
Definition 3.6. Let char(k) = 2, 3, let J be a Jordan k-algebra, let X be a special J-module with action •.
Theorem 3.7. Let char(k) = 2, 3. Let J be the reduced spin factor of the non-degenerate anisotropic quadratic space (k, V, q) with base point u. Let X be a non-trivial J-ternary algebra such that ( ( (u•x, x) ) ) = 0 for all x ∈ X 0 \{0}. Then X satisfies (3.4). Therefore (k, V, q, u, X 0 , ·, h) is a quadrangular algebra, with · and h as in Theorem 3.5.
Since this last equation holds we have proved that (3.4) holds.
(ii) We have to demand that ( ( (x, x, x) ) ) = 0 for all x ∈ X 0 \ {0}, because there exist J-ternary algebras which fulfill all the requirements but where ( ( (x, x, x) ) ) = 0 for some x = 0 ∈ X 0 . For example, consider [ABG,Example 6.81] with the zero skew hermitian form. Examples like this we clearly want to avoid.

Construction of quadrangular algebras of pseudo-quadratic form type
Let k be a field of characteristic not 2.
We rely on the example [ABG,6.81] to obtain a quadrangular algebra of pseudo-quadratic form type using Theorem 3.5. In combination with Section 3.4 this will show that all quadrangular algebras of characteristic not 2 can be obtained using the construction in Theorem 3.5.
In Section 4 we will show that each Moufang quadrangle in characteristic not 2 can be obtained from a construction that generalizes the construction in Theorem 3.5.
Definition 3.9. (i) Define J = H(M 2 (L), σ) (see Definition 2.21 and Remark 2.22); this Jordan algebra is a reduced spin factor of the quadratic form As before we define e 0 = 1 0 0 0 , e 1 = 0 0 0 1 , u = 0 1 1 0 ∈ J : u is a base point of Q. (ii) Define X = X 2 , the 1 × 2 row vectors over X. (iii) We define the action of J on X as 7 j • x := xj ∈ X for j ∈ J, x ∈ X.
Under the identifications the quadrangular algebra defined in Theorem 3.5 is exactly the pseudo-quadratic space we started with: (k, L, q, u, X, scalar multiplication, h) Proof. Verifying that X is a special J-module and (3.1) requires some straightforward calculations. We will verify (3.4) and (3.5). Define Hence ( ( (u• x, x) ) ) is equal to 0 if and only if h(x, x) = 0. As we are working in a standard anisotropic pseudo-quadratic space, π(x) = 1 2 h(x, x) is anisotropic (see Remark 2.11.2), so (3.5) holds.

Condition (3.4) holds since
From Theorem 3.5 we conclude that (k, J 1 / 2 , Q, u, X 0 , ·, h) is a quadrangular algebra with 3.4 Construction of quadrangular algebras of type E 6 , E 7 , E 8 In this subsection we give a new construction of the vector spaces X, L and the maps · and h that we discussed in Section 2.3.2.

A characterization of quadratic forms of type E 6 , E 7 , E 8
We start by giving a new way to describe quadratic forms of type E 6 , E 7 and E 8 (see Definition 2.15). The following illuminating observation was made by Skip Garibaldi.
Theorem 3.11. Let q be an anisotropic form over k of dimension 6, 8 or 12. Then q is of type E 6 , E 7 or E 8 respectively if and only if there exist an octonion division algebra C 1 and a division composition algebra C 2 , of dimension 2, 4 or 8 respectively, that have linkage number one such that q is similar to the anisotropic part of the Albert form of C 1 ⊗ k C 2 .
The 'only if'-direction of this theorem is proved in the following, more technical, lemma.
Lemma 3.12. We consider a quadratic form q of type E 6 , E 7 or E 8 . Let N denote the norm of a separable quadratic field extension E = k(x)/(x 2 − a) for a / ∈ k 2 ; (i) If q = N ⊗ 1, s 2 , s 3 of type E 6 , define C 1 = (a, −s 2 , −s 3 ) k and C 2 = E.
Then q is similar to the anisotropic part of the Albert form of C 1 ⊗ k C 2 and C 1 and C 2 are division algebras that have linkage number 1.
Proof. Denote the norm form of C 1 by q 1 , the norm form of C 2 by q 2 and the Albert form of C 1 ⊗ k C 2 by q A . In the case that q is of type E 8 we will verify that the other two cases are similar. We have q 1 = N ⊗ s 2 , s 3 and q 2 = N ⊗ s 4 s 6 , s 5 s 6 . Since tH ≃ H for t ∈ k it follows that s 2 s 3 ( 1, s 2 , s 3 , s 4 , s 5 , s 6 ⊥ H) ≃ s 2 s 3 s 3 , s 2 , 1, s 5 , s 4 , s 6 ⊥ H ≃ s 2 , s 3 , s 2 s 3 , s 2 s 3 s 5 , s 2 s 3 s 4 , s 2 s 3 s 6 ⊥ H ≃ s 2 , s 3 , s 2 s 3 , −s 4 s 6 , −s 5 s 6 , −s 4 s 5 ⊥ 1, −1 ≃ s 2 , s 3 ⊥ − s 4 s 6 , s 5 s 6 .
(3.8) By multiplying the above identity with N we obtain (3.7).
Note that q is anisotropic. Therefore q 1 ⊥ −q 2 has Witt index 2; it follows that q 1 and q 2 are anisotropic and both C 1 and C 2 are division algebras. It follows from (3.7) that q A has Witt index 1, and now Lemma 2.7 implies that C 1 and C 2 have linkage number 1.
Proof of Theorem 3.11. The 'only if'-direction is proven in the Lemma above. The 'if'-direction follows in a similar way. We elaborate the case where C 1 and C 2 are octonion division algebras.
Since C 1 and C 2 contain an isomorphic field extension, by [SV,Prop. 1.5.1] we can assume that C 1 = (a, b 1 , c 1 ) k and C 2 = (a, b 2 , c 2 ) k for some a, b 1 , b 2 , c 1 , c 2 ∈ k. We denote the Albert form of C 1 ⊗ k C 2 by q A .
Define N := −a , this is anisotropic since C 1 is division. By going through (3.8) from bottom to top with we find that q A is similar to N ⊗ 1, s 2 , s 3 , s 4 , s 5 , s 6 ⊥ H. Since the Witt index of q A is one, N ⊗ 1, s 2 , s 3 , s 4 , s 5 , s 6 is the anisotropic part of q A ; since s 2 s 3 s 4 s 5 s 6 = −1 it is of type E 8 .

The construction
In order to construct quadrangular algebras of type E 6 , E 7 and E 8 we follow Example 6.82 in [ABG] closely. In loc. cit. a J-ternary algebra is constructed out of C 1 ⊗ k C 2 in characteristic zero, but this restriction is not necessary.
First we give a motivation of the approach we will be following in our construction.
Remark 3.13. Let C 1 be an octonion division algebra and C 2 a separable quadratic field extension, quaternion division algebra or octonion division algebra and assume that C 1 and C 2 have linkage number one.
The dimension of C 1 ⊗ k C 2 is 16, 32 or 64, respectively. The space of skew elements is S = S 1 ⊗ 1 + 1 ⊗ S 2 and has dimension 8, 10 or 12, respectively. Let (k, L, q, 1, X , ·, h) be a quadrangular algebra of type E 6 , E 7 or E 8 , respectively. We summarize some dimensions: We see that in all three cases dim k S = dim k L + 2 and dim k (C 1 ⊗ C 2 ) = 2 dim k X.
In Theorem 3.5 we considered some objects the dimensions of which behave similarly: Let J be a Jordan algebra of reduced spin type with base point and let X be a special J-module. Then dim k (J) = dim J 1 / 2 + 2 and dim k X = 2 dim k X 0 . From Lemma 3.12, the Albert form from S to k can be written as the sum of a hyperbolic plane and a quadratic form of type E 6 , E 7 or E 8 , respectively. Note that a hyperbolic plane is two-dimensional.
In the following pages, we will give S the structure of a reduced spin factor of a quadratic form of type E 6 , E 7 or E 8 , respectively, and identify J 1 / 2 with L. Then we will give C 1 ⊗ C 2 the structure of a special J-module equipped with a bilinear skew-hermitian form, and identify (C 1 ⊗ C 2 ) 0 with X.
We start by fixing some notation.
Notation 3.14. (i) We fix a basis for the composition algebras C 1 and C 2 that have linkage number 1. We let C 1 be the octonion division algebra If C 2 is a separable quadratic field extension, we define C 2 = 1, i 2 . In the case C 2 is a quaternion division algebra we define C 2 = 1, i 2 , j 2 , i 2 j 2 .
In the case C 2 is an octonion division algebra we define Since C 1 and C 2 have linkage number 1, we can choose these bases in such a way that i 2 1 = i 2 2 =: a ∈ K. (ii) From now on we denote S 1 , S 2 and S for the set of skew elements of C 1 , C 2 and C 1 ⊗ k C 2 , respectively. (iii) We denote the Albert form of C 1 ⊗ k C 2 by q A : S → k and its associated bilinear form by f A . (iv) Let V := i 1 ⊗ 1, 1 ⊗ i 2 ⊥ denote the orthogonal complement of the subspace i 1 ⊗1, 1⊗i 2 of S with respect to the non-degenerate bilinear form f A .
We want to make S into a Jordan algebra of reduced spin type. In particular it should contain supplementary proper idempotents e 0 and e 1 and an element u ∈ J 1 / 2 such that u 2 is the identity. It will become clear that the elements constructed in the following lemma will be the ones we need.
Lemma 3.15. Let u ∈ V \ {0} be arbitrary. Then up to scalars, there exists a unique pair (e 0 , e 1 ) of elements in S such that Explicitly, there exists an element λ ∈ k such that (λe 0 , λ −1 e 1 ) is equal to Proof. Since q A has Witt index one, q A is anisotropic on V = i 1 ⊗1, 1⊗i 2 ⊥ . Hence q A (u) = 0.
Since dim S = dim V + 2, we want to make V into a quadratic space. If we want that u.u = 1 in the Jordan algebra of reduced spin type we will define, the element u should be the base point of the quadratic form that determines the reduced spin factor. In the following definition we define a Jordan algebra on S; in Lemma 3.17 we will show that this Jordan algebra has a natural interpretation in the endomorphism ring of C 1 ⊗ k C 2 .
Definition 3.16. Let u ∈ V \ {0} and (i) We define a quadratic form on the vector space V , We denote the corresponding bilinear form by F . It follows from Theorem 3.11 that (k, V, Q) is a quadratic space of type E 6 , E 7 or E 8 , respectively, with base point u. (ii) We have S = ke 0 ⊕ V ⊕ ke 1 , we define the Jordan multiplication as in Definition 2.20: We denote this Jordan algebra by J, this is the reduced spin factor of (k, V, Q).
where the inverse and ♮ is as in Definition 2.3. Notice that e 0 +e 1 is the identity in the Jordan algebra J on S, the definition of r has nothing to do with the inverse in J.
(iv) Let s ∈ S, define L s ∈ End k (C 1 ⊗ k C 2 ) as L s x := sx for all x ∈ C 1 ⊗ k C 2 .
We will consider the Jordan algebra of the associative algebra End k (C 1 ⊗ k C 2 ), denoted by End k (C 1 ⊗ k C 2 ) + . We show that the algebra of reduced spin type we defined above, is isomorphic to a Jordan subalgebra of End k (C 1 ⊗ k C 2 ) + .
Therefore L S L r is a Jordan subalgebra of End k (C 1 ⊗ k C 2 ) + isomorphic to J.
Proof. We will make use of [A3,Proposition 3.3 (3.8)]. In [A3] only characteristic 0 is considered; however this proposition can be generalized to characteristic different from 2 without any adjustments. The proof of this proposition uses basic identities of octonions (see Lemma 2.1) and the identity s 1 (s 2 (s 1 x)) = (s 1 s 2 s 1 )x for x ∈ C 1 ⊗ k C 2 (see Lemma 2.4).
In order to define an action of J on C 1 ⊗ k C 2 , we use the isomorphism of the previous Lemma.
Definition 3.18. We define the bilinear action We define the skew symmetric bilinear map ( ( (., .) ) ) : Remark 3.19. (i) After some computation we find for all x 1 ∈ C 1 , x 2 ∈ C 2 . Note that this is independent of the choice of the base point u. (ii) Let C be a composition algebra and define the bilinear skew symmetric map ψ : C × C → S : (x, y) → xy − yx.
Then for all x 1 , y 1 ∈ C 1 , x 2 , y 2 ∈ C 2 we have, In the following theorem we show that the construction given in the introduction does indeed give rise to a quadrangular algebra of type E 6 , E 7 or E 8 . In the proof we make a distinction between the cases char(k) = 2 and char(k) = 2, 3. When char(k) = 2, 3, C 1 ⊗ k C 2 is a structurable algebra and we can use the theory of structurable algebras.
If char(k) = 3 we can not make use of the theory of structurable algebras; therefore we prove this in a direct way only making use of identities in octonions. Regrettably, this gives rise to lengthy computations and for one particular identity we had to rely on the computer algebra software [Sage]. This proof does not use the fact that the characteristic is equal to 3, but only that it is different from 2.
Theorem 3.20. Let char(k) = 2. Let e 0 , e 1 , u ∈ S be as in Lemma 3.15, let the quadratic form Q of type E 6 , E 7 , E 8 and the reduced spin factor J be as in Definition 3.16. Let X := C 1 ⊗ k C 2 , let • and ( ( (., .) ) ) be defined as above.
Then (k, V, Q, u, X 0 , ·, h) is a quadrangular algebra of type E 6 , E 7 , E 8 . Proof. We have from Lemma 2.4, that (e 0 + e 1 ) • x = x for all x ∈ X. The fact that X is a special J-module now follows from Lemma 3.17. It follows from Theorem 2.12 that if (k, V, Q, u, X 0 , ·, h) is a quadrangular algebra, it has to be of type E 6 , E 7 , E 8 due to the dimension of V .
In [ABG,Remark 6.7] it is pointed out that each structurable algebra A (in our case C 1 ⊗ k C 2 ) with an invertible skew element is a J-ternary algebra with J = L S L r ⊂ End k (A) + . The action of the Jordan algebra on X and the skew bilinear map are defined as in Definition 3.18 above; the trilinear product is defined as X × X × X → X : (x, y, z) → −V x,ry z := (x(yr))z + (z(yr))x + (zx)(ry).
[ABG] only considers fields of characteristic 0. We checked that every structurable algebra, with an invertible skew element, is a J-ternary algebra, in characteristic different from 2 and 3. This proof is omitted in [ABG,Remark 6.7] and uses deep identities in structurable algebras. We thank Bruce Allison for giving us a detailed explanation on how to prove this fact.
For the proof of (3.5) we refer to the general characteristic case below. It now follows from Theorem 3.7 that (k, V, Q, u, X 0 , ·, h) is a quadrangular algebra.

char(k) = 2
We first verify that the identity in the right hand side of (3.1) holds, this takes a rather lengthy but straightforward computation: Since the condition is linear in x and y, one can choose x = x 1 ⊗ x 2 and y = y 1 ⊗ y 2 for x 1 , y 1 ∈ C 1 , x 2 , y 2 ∈ C 2 . Let s = s 1 ⊗ 1 + 1 ⊗ s 2 ∈ S and denote r = r 1 ⊗ 1 + 1 ⊗ r 2 , instead of using its definition with coordinates. Using Remark 3.19.(ii) it is not hard to show that the following identities hold for i ∈ {1, 2} Using these identities, (3.1) can be simplified to and this identity can be checked using Lemma 2.1, especially the Moufang identities (v).
We were not able to verify (3.4) by hand. The problem is that (3.4) has degree 3 in x, so we can not assume that x is of the form e 0 • (x 1 ⊗ x 2 ). We did a computation based on a coordinatization of X, we used the software [Sage] to do the symbolic computations. Now x is an arbitrary element in X 0 = e 0 • X, therefore x is a sum of elements of the form x 1 ⊗ x 2 + 1 a i 1 x 1 ⊗ i 2 x 2 (see Remark 3.19). We implemented octonions and the tensor product of two octonions in Sage in a symbolic way, and we verified that (3.4) holds.
The only fact that remains to be verified is (3.5). In fact, this is exactly axiom (D2) and in the proof of Theorem 3.5 the condition (3.5) is not used to prove any of the other axioms. Since we already know that the axioms A-B-C-D1 are true, we will use these to prove ( ( (u•x, x) ) ) = 0 for all x ∈ X 0 \{0}.

First we show that
there exists an x ∈ X 0 such that ( ( (u • x, x) ) ) = 0.
The rest of the proof is inspired by the proof given in [TW,Theorem 13.47].
We fix an arbitrary x = 0 ∈ X 0 , notice that we no longer assume that x has the form e 0 • (x 1 ⊗ x 2 ). We suppose that ( ( (u • x, x) ) ) = 0 and aim to get a contradiction. It follows 8 from (3.4) that ( ( (v • x, x) ) ) = 0 for all v ∈ V .
From now on we assume that y ∈ X 0 is such that ( ( (u • x, y) ) ) = 0. Next we show that (3.10) Since this identity is trivial for u = v, we assume v ⊥ u. Then (3.10) is equivalent to (3.11) We consider (3.4) for y + tx for a parameter t ∈ k, we compare the terms that have degree one in t using the assumption that ( ( (v • x, x) ) ) = 0 for all v ∈ V , and we get since v( ( (x, y) ) ) ∈ J 1 and x ∈ X 0 . This proves (3.11).
From (3.10) we have (x · ( ( (u • x, y) ) )) · V ⊆ x · V ; since ( ( (u • x, y) ) ) = 0, the dimension of those two vector spaces is equal and we find that (3.12) For arbitrary w ∈ V it follows from (3.10) that From (3.12) we find that (x · V ) · V = x · V and hence x · C(q, u) = x · V = X 0 contradicting the irreducibility of X 0 . This finishes the proof of Theorem 3.20.
It follows from the previous theorem that we have, in characteristic not 2, a new coordinate-free definition of the various maps introduced in [TW,Chapter 13].
Remark 3.22. The reader might wonder what will happen if we apply our construction in the case that both C 1 and C 2 are composition algebras of dimension 2 or 4 with mutual linkage number 1. In the three different cases that arise in this way, we get the following dimensions for the different relevant vector spaces.
In the first case, the vector space L is trivial, so our construction no longer applies (we cannot find an element u ∈ V \ {0} needed in Definition 3.16).
In the second case, the spaceX gets the structure of a 2-dimensional vector space over E, and the corresponding quadrangular algebra is isomorphic to a quadrangular algebra of pseudo-quadratic form type with underlying vector spaceX. Notice that E ⊗ Q ∼ = M 2 (E) since E and Q are 1-linked.
Similarly, in the third case, the spaceX gets the structure of a 2dimensional vector space over a quaternion division algebra Q 3 , and the corresponding quadrangular algebra is isomorphic to a quadrangular algebra of pseudo-quadratic form type with underlying vector spaceX. The algebra Q 3 is the quaternion algebra with norm form similar to q A , and in this case Q 1 ⊗ Q 2 ∼ = M 2 (Q 3 ).

A final remark
In an earlier paper [BD], we had found another related but quite different class of structurable algebras that seems to play an important role in the understanding of the exceptional Moufang quadrangles of type E 6 . E 7 and E 8 . That structurable algebra is, in each case, obtained by doubling another algebra (instead of halving an algebra as we did in the current paper).
More precisely, to each Moufang quadrangle Ω of type E 6 , E 7 or E 8 , we can associate a structurable algebra Y , the isotopy class of which is a complete invariant of the Moufang quadrangle Ω, and which is obtained by applying the so-called Cayley-Dickson doubling process on the Jordan algebra A + , where (i) A is a quaternion algebra Q if Ω is of type E 6 ; (ii) A is a tensor product Q ⊗ E with Q a quaternion algebra and E a quadratic extension, if Ω is of type E 7 ; (iii) A is a biquaternion algebra Q 1 ⊗ Q 2 if Ω is of type E 8 . However, we are not yet aware of a direct way of relating the structurable algebra Y with the structurable algebra X = C 1 ⊗ C 2 which we have investigated in the current paper.

A unified construction for Moufang quadrangles
in characteristic not 2

Preliminaries on Moufang quadrangles
A Moufang polygon is a notion from incidence geometry introduced by Jacques Tits. We only give a brief summary of the theory of Moufang quadrangles, and we refer to [TW] for more details. The importance will immediately become clear in Theorem 4.1 below.
A generalized quadrangle Γ is a connected bipartite graph with diameter 4 and girth 8. We call a generalized polygon thick if every vertex has at least 3 neighbors. A root in Γ is a (non-stammering) path of length 4 in Γ.
Let Γ be a thick generalized quadrangle, and let α = (x 0 , . . . , x 4 ) be a root of Γ. Then the group U α of all automorphisms of Γ fixing all neighbors of x 1 , x 2 , x 3 (called a root group) acts freely on the set of vertices incident with x 0 but different from x 1 . If U α acts transitively on this set (and hence regularly), then we say that α is a Moufang root.
A Moufang quadrangle is a generalized quadrangle for which every root is Moufang. We then also say that Γ satisfies the Moufang condition.
Moufang quadrangles have been classified by J. Tits and R. Weiss [TW]. Loosely speaking, the result is the following.
Theorem 4.1 ( [TW]). Every Moufang quadrangle arises from an absolutely simple linear algebraic group of relative rank 2, or from a corresponding classical group or group of mixed type.
In particular, every Moufang quadrangle is of "algebraic origin", and in fact, the Moufang quadrangles provide a useful tool to help in the understanding of the corresponding groups; this is particularly true for the Moufang quadrangles arising from linear algebraic groups of exceptional type. For instance, the Kneser-Tits problem for groups of type E 66 8,2 has recently been solved using the theory of Moufang polygons [PTW].
In order to describe a Moufang quadrangle in terms of algebraic data, we will use so-called root group sequences. A root group sequence for a Moufang quadrangle is a sequence of 4 root groups, labeled U 1 , . . . , U 4 , together with commutator relations describing how elements of two different root groups U i and U j commute. In each case, the commutator of an element of U i and U j (with i < j) belongs to the group U i+1 , . . . , U j−1 . The following result is crucial.
Theorem 4.2. Let Γ be a Moufang quadrangle. Then Γ is completely determined by the root groups U 1 , . . . , U 4 together with their commutator relations.
For more details about this procedure, and how the Moufang polygons can be reconstructed from the root group sequences, we refer to [TW] or to the survey article [DV].
For each type of Moufang quadrangle, we will describe an algebraic structure which will allow us to parametrize the root groups and describe the commutator relations.
In principle, it is possible to define a single algebraic structure to describe all possible Moufang quadrangles; this gives rise to the so-called quadrangular systems which have been introduced by the second author [D1]. These structures, however, have some disadvantages from an algebraic point of view; most notably, the definition does not mention an underlying field of definition (although it is possible to construct such a field from the data), and the axiom system looks very wild and complicated, with no less than 20 defining identities.
The Moufang quadrangles of types (2)-(4) are often called classical, those of type (5) and (6) are called exceptional and those of type (1) are of mixed type. Since the Moufang quadrangles of type (1) and (6) only exist over fields of characteristic two, and moreover are not directly related to rank two forms of algebraic groups, we exclude those two classes from our further discussion.
In the following section we will give a uniform description of the remaining 4 classes of Moufang quadrangles, over fields of characteristic different from 2, starting from a special Jordan module.

Construction of Moufang quadrangles from special Jordan modules
We will show that each type of Moufang quadrangle in characteristic not 2 can be described in a unified way from a special J-module. We generalize the procedure that we used in Theorem 3.5 to obtain quadrangular algebras.
In order to obtain all Moufang quadrangles we allow that dim(J 0 ) > 1 and we allow the special J-module to be the trivial module. It follows from Theorem 4.2 that it is sufficient to describe the 4 root groups and the commutator relations of the root groups to describe the Moufang quadrangle completely.
Construction 4.3. Let J be a non-degenerate Jordan algebra that contains supplementary proper idempotents e 0 and e 1 . Let J 0 , J 1 / 2 , J 1 be the Peirce subspaces of J with respect to e 1 . We assume that each element in J 1 / 2 \ {0} is invertible and that there exists u ∈ J 1 / 2 such that u 2 = 1.
Let U 1 and U 3 be two groups isomorphic to W , and let U 2 and U 4 be two groups isomorphic to V . Denote the corresponding isomorphisms by we say that U 1 and U 3 are parametrized by W and that U 2 and U 4 are parametrized by V . Now, we implicitly define the group U + = U 1 , U 2 , U 3 , U 4 by the following commutator relations: • a 1 , a 2 ) ) ) , It follows from Lemma 2.23 that J should be either of reduced spin type or of type H(M 2 (L), σt). For each of these two cases, we will distinguish between the zero special J-module and a non-zero J-module. Case by case, we will show that in this way the root groups U 1 , U 2 , U 3 , U 4 and commutation relations given above coincide with the description given in Chapter 16 of [TW] of the Moufang quadrangles in characteristic not 2.
Remark 4.4. In [D2] quadrangular systems are introduced. These are structures that are defined by 24 axioms, which describe in a unified way all Moufang quadrangles (including characteristic 2.) We believe it should be possible to start with Construction 4.3, impose a few more axioms that look like the ones in Theorem 3.5 and prove all the axioms defining a quadrangular system. However the verifications of the axioms that use the map κ, this is a kind of "multiplicative inverse" in the group W , get very complicated.
Moufang quadrangles of quadratic form type Let J be a reduced spin factor of an anisotropic, non-degenerate quadratic space (k, V, q) with base point u. Let X be the zero module over J.
Let U 1 and U 3 be parametrized by W and U 2 and U 4 be parametrized by V . Let t, t 1 , t 2 ∈ k, v, v 1 , v 2 ∈ V ; using the formulas for the multiplication and the U -operator in a Jordan algebra of reduced spin type (see Definition 2.20) we find for the commutator relations [U i , U i+1 ] = 1 ∀i ∈ {1, 2, 3} .
We obtain exactly the same description as in [TW,Example 16.3].
If d = dim K V is finite, then these Moufang quadrangles arise from linear algebraic groups; they are of absolute type B ℓ+2 if d = 2ℓ + 1 is odd, and of type D ℓ+2 if d = 2ℓ is even.
Moufang quadrangles of type E 6 , E 7 , E 8 This case was actually already handled in Theorem 3.20, since from quadrangular algebras one can define the root groups and commutation relations of the corresponding Moufang quadrangles, see [W,Chapter 11]. Now we quickly verify that we get indeed the right commutator relations using Construction 4.3.
Let J be a reduced spin factor of an anisotropic, non-degenerate quadratic space (k, V, q) with base point u, let X = C 1 ⊗ k C 2 and let the skewsymmetric form ( ( (., .) ) ) be as in Section 3.4. Since quadrangular algebras of type E 6 , E 7 and E 8 are determined by the similarity class of their there quadratic space. We have that the following maps coincide with the maps defined in [TW,Chapter 13]: • a, b) ) ), g(a, b)e 0 = 1 2 ( ( (b, a) ) ).
We obtain exactly the same description as in [TW,Example 16.6]. The Tits indices of the corresponding linear algebraic groups are as follows.
Moufang quadrangles of pseudo-quadratic type These are obtained in similar fashion as the quadrangular algebras in Section 3.3, but here we start from an arbitrary skew field with involution instead of starting from a quadratic pair. We repeat part of the setup from Section 3.3.
Let L be a skew-field with involution σ. Let (L, σ, X, h, π) be a standard pseudo-quadratic space (see Section 2.3.1), so π(a) = 1 2 h(a, a) for all a ∈ X. Let J = H(M 2 (L), σt) (see Definition 2.21) and let X = X 2 , the 1 × 2 row vectors over X. For the action of J on X, for j ∈ J, a ∈ X we have j • a = aj ∈ X.
When we translate the group law and commutator relations in [TW,Example 16.5] from T to X × L σ using φ, we indeed obtain the expressions written above.
If L is finite-dimensional over its center, of degree d, and X is finitedimensional over L, then these Moufang quadrangles arise from algebraic groups. If the involution is of the second kind, they are outer forms of absolute type A ℓ . If the involution is of the first kind, they are of absolute type C ℓ or D ℓ .