Involutions of a Clifford algebra induced by involutions of orthogonal group in characteristic 2

M. G. Mahmoudi, A.-H. Nokhodkar

September 26, 2012

Abstract Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences and their decompositions are investigated.

Mathematics Subject Classification: 11E88, 11E04, 16W10.

1 Introduction

The main object of this work is to study the involutions of a Clifford algebra induced by the involutions of the orthogonal group in characteristic two. For the case of characteristic ≠ 2, these involutions were studied in [8] in connection with the periodicity of real Clifford algebras with involution. Also they were considered in [15] in connection with the Pfister Factor Conjecture, which was finally settled in [1]. In characteristic ≠ 2, some properties of these involutions were also investigated in [9].

Our starting point is a general analysis of involutions of orthogonal group in characteristic 2. For the case of characteristic different from 2, the situation is much more straightforward as it is known that if τ is an involution in the orthogonal group O(V, q) then there exists a decomposition V = U ⊥ U^{⊥} with τ|_{U} = id and τ|_{U^{⊥}} = − id. For the case where the characteristic of the base field is 2, the situation is quite more subtle. Wiitala in [17, Thm. 2] provides a characterization of involutions of orthogonal groups in characteristic 2. For every involution τ in O(V, q), he proves that there exists a decomposition V = W ⊥ V_{1} ⊥ V_{2} ⊥ ⋯, where the restriction of τ to W is trivial and exactly one of the following is true: (1) each V_{i} is a two-dimensional subspace of V and the restriction of τ to V_{i} is a reflection; (2) each V_{i} is a four-dimensional subspace of V invariant under τ such that the fixed space of τ|_{V_{i}} is a totally isotropic space of dimension 2. In §3, several geometric interpretations of Wiitala’s result are obtained and his characterization is complemented.

An initial observation for the study of the involutions is that every involution of a Clifford algebra C(V) which leaves V invariant, is induced by an involution in the orthogonal group O(V, q). Let τ ∈ O(V, q) be an involution and let J_{τ} be the involution induced by τ on C(V). In §4, elementary investigations about these involutions such as the calculation of their type (that is if J_{τ} is of orthogonal or symplectic type) are done. Also some relations between invariants of J_{τ} (e.g., the type and the discriminant (in the sense of [7])) and invariants of τ (e.g., the Dickson invariant and the spinor norm) are obtained. Finally, in the case of the nontriviality of the Arf invariant, these questions for the restriction of J_{τ} to the even Clifford algebra C_{0}(V) are investigated.

In §5, involutorial versions of some relations between the Clifford algebra and the even Clifford algebra (needed in the sequel) obtained by Mammone, Tignol and Wadsworth in [10] are presented. These relations link an orthogonal sum of a quadratic plane (i.e., a two-dimensional quadratic space) over F and an n-dimensional quadratic space over F to the Clifford algebra of an (n + 2)- dimensional quadratic space.

In §6, analogous to [9], a geometric characterization of an involution of a quaternion algebra is obtained. It turns out that if (Q, σ) is a quaternion algebra with involution then there exists a quadratic plane (𝔼, φ) and an involution τ ∈ O(𝔼, φ) such that (Q, σ) ≃ (C(𝔼), J_{τ}). Also σ is of symplectic (resp. orthogonal) type if and only if τ = id (resp. τ is a reflection), see (6.1). In view of this result, it is deduced that as an algebra with involution, (Q, σ) is characterized by the spinor norm of τ. Then we show that a multiquaternion algebra with involution (A, σ) := (Q_{1}, σ_{1}) ⊗ ⋯ ⊗ (Q_{n}, σ_{n}) is isomorphic to the Clifford algebra with involution (C(V), J_{τ}) where τ ∈ O(V, q) is a suitable involution of a certain quadratic space (V, q). Also a more precise description of τ, depending on the type of σ, is given. In particular it is shown that if σ is of symplectic type then the involution τ ∈ O(V, q) can be chosen to be identity and if σ is of orthogonal type then τ can be chosen to be an orthogonal sum of reflections (see (6.3)).

Let (Q, σ) be a quaternion algebra with an involution of the second kind. In contrary to what happens in characteristic different from 2 (see [9, (6.10)]), (Q, σ) is not isomorphic to a Clifford algebra with involution (C(V), J_{τ}) for any (V, q) and τ (6.9). Instead of that it is shown that (Q, σ) is isomorphic to an even Clifford algebra with involution (C_{0}(V), J_{τ}) where (V, q) is a quadratic space of dimension 4 over the fixed field of σ restricted to the center of Q. More generally if K/F is a separable quadratic extension with nontrivial automorphism ρ, it is shown that the multiquaternion algebra (A, σ) := (K, ρ) ⊗ (Q_{1}, σ_{1}) ⊗ ⋯ ⊗ (Q_{n}, σ_{n}) is isomorphic to the even Clifford algebra with involution (C_{0}(V), J_{τ}) for a suitable quadratic space (V, q) over F and an involution τ in O(V, q). Furthermore it is shown that if n ≡ 0 mod 2 then τ can be chosen to be identity and if n ≡ 1 mod 2 then τ can be chosen to be a reflection (see (6.7)). In should be mentioned that in low dimensions (e.g., n ⩽ 2), these results, for arbitrary characteristic, are already included in [5].

The main object of §7 is to investigate the behavior of the involution J_{τ} on C_{0}(V) in the case where (V, q) has trivial Arf invariant. Note that as in this case C_{0}(V) is not central simple, one may wonder how does J_{τ} act on simple components of C_{0}(V). Also one may want to see what is the relation between the spinor norm of τ and the discriminant of the restriction of J_{τ} to simple components of C_{0}(V). In this section these questions and some other related questions similar to those investigated in the previous sections, are studied.

It is relevant to mention that in characteristic ≠ 2, the behavior of J_{τ} on simple components of C_{0}(V), even for the simplest case τ = id, has been of importance in many applications of Clifford algebras in the literature (see, e.g., [3], [16], [11], [13]).

In characteristic 2, when τ is arbitrary we are able to give a precise description of the action of J_{τ} on simple components of C_{0}(V). It turns out that, depending on τ, the involution J_{τ} can also be a switch involution. See (7.8) and (7.11).

In a final appendix, a universal mapping property for the even Clifford algebra, analogous to [9, (3.3)], which is used to shorten some proofs in the paper is presented.

2 Preliminaries and terminology

Let F be a field of characteristic 2 and let V be a finite dimensional vector space over F. A quadratic form over F is a map q : V → F such that q(λu + μv) = λ^{2}q(u) + μ^{2}q(v) + λμB(u, v) for every λ, μ ∈ F and u, v ∈ V, where B : V × V → F is a bilinear form over V. We have B(u, v) = q(u) + q(v) + q(u + v) and so the bilinear form B is uniquely determined by q. In particular B(v, v) = 0 for every v ∈ V, i.e., B is an alternating form. Two vectors u, v ∈ V are called orthogonal if B(u, v) = 0. The radical of V is defined as rad V = {x ∈ V | B(x, y) = 0 for every y ∈ V }. We say that q is nondefective if ker(q|_{radV}) = 0. If the bilinear form B is nondegenerate, (V, q) is called regular. We call (V, q) or (V, q, B) a quadratic space if q is regular. Any quadratic space (V, q) is necessarily even dimensional and has a basis B = {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} such that B(u_{i}, v_{i}) = 1 for each i and all other distinct pairs of B are orthogonal (see [14, p. 339]). Such a basis is called a symplectic basis of V. We denote by [a, b] the two dimensional quadratic space (V, q) where V = F u + F v, q(u) = a, q(v) = b and B(u, v) = 1. Also we call {u, v} a standard symplectic basis of (V, q).

We say that q represents a ∈ F, if there is a nonzero vector v ∈ V such that q(v) = a. The set of values in F^{×} = F \ {0} represented by q is denoted by D_{F} (q). The quadratic space (V, q) is called universal if D_{F} (q) = F^{×}. If (V, q, B) is a quadratic space and a ∈ F^{×}, the scaled quadratic space (V, a · q, a · B) is defined by a · q(v) = aq(v) for every v ∈ V.

An isometry between two quadratic spaces such as (V_{1}, q_{1}) ≃ (V_{2}, q_{2}) is an isomorphism of vector spaces τ : V_{1} → V_{2} such that q_{2}(τ(v)) = q_{1}(v) for every v ∈ V_{1}. Similarly, an isometry between two bilinear spaces such as (V_{1}, B_{1}) ≃ (V_{2}, B_{2}) is an isomorphism of vector spaces τ : V_{1} → V_{2} such that B_{2}(τ(u), τ(v)) = B_{1}(u, v) for every u, v ∈ V_{1}. Note that (V_{1}, q_{1}) ≃ (V_{2}, q_{2}) implies that (V_{1}, B_{1}) ≃ (V_{2}, B_{2}) but the converse is not always true. The group of isometries of (V, q) is denoted by O(V, q) and is called the orthogonal group of (V, q).

Let (V_{1}, q_{1}) and (V_{2}, q_{2}) be two quadratic spaces over F and define the quadratic form q_{1} ⊥ q_{2} over V_{1} ⊕ V_{2} via (q_{1} ⊥ q_{2})(v_{1} + v_{2}) = q_{1}(v_{1}) + q_{2}(v_{2}). We denote this quadratic space by (V_{1} ⊥ V_{2}, q_{1} ⊥ q_{2}) and we call it the orthogonal sum of (V_{1}, q_{1}) and (V_{2}, q_{2}). If (V_{1}, q_{1}) and (V_{2}, q_{2}) are quadratic spaces over F and τ_{1} and τ_{2} are isometries of (V_{1}, q_{1}) and (V_{2}, q_{2}), respectively, the orthogonal sum of τ_{1} and τ_{2} which is denoted by τ_{1} ⊥ τ_{2} is an isometry of (V_{1} ⊥ V_{2}, q_{1} ⊥ q_{2}) defined by (τ_{1} ⊥ τ_{2})(v_{1} + v_{2}) = τ_{1}(v_{1}) + τ_{2}(v_{2}).

All F -algebras considered in this work are implicitly supposed to be associative. The center of an F -algebra A is denoted by Z(A) and the group of invertible elements of A is denoted by A^{∗}. If u ∈ A is an invertible element, we use the notation Int(u) for the inner automorphism of A induced by u:

Int(u) : A → A, Int(u)(x) = uxu^{−1}.

Let (V, q) be a quadratic space over a field F of characteristic 2 and let A be an F -algebra. A linear map f : V → A is called compatible with q if for every x ∈ V, we have f(x)^{2} = q(x) · 1. In particular if {v_{1}, ⋯, v_{n}} is a basis of V, then the linear map f is compatible with q if and only if for every 1 ⩽ i ⩽ n, we have f(v_{i})^{2} = q(v_{i}) · 1 and for every 1 ⩽ i, j ⩽ n, we have f(v_{i})f(v_{j}) + f(v_{j})f(v_{i}) = B(v_{i}, v_{j}) · 1.

The Clifford algebra of a quadratic space (V, q) is denoted by C(V, q), or C(V) if no confusion arises, and we consider V as a subspace of C(V). We have uv = vu + B(u, v) · 1 for every u, v ∈ V. The even Clifford algebra of (V, q) is denoted by C_{0}(V, q) or C_{0}(V).

If τ is an isometry of (V, q), then τ induces an algebra automorphism I_{τ} of C_{0}(V). The Dickson invariant

D : O(V, q) → ℤ/2ℤ

is defined by D(τ) = 0 ∈ ℤ/2ℤ, if I_{τ} |_{Z(C_{0}(V))} = id and D(τ) = 1 ∈ ℤ/2ℤ otherwise.

Consider the additive subgroup ℘(F) = {x + x^{2} |x ∈ F } of (F, +). Let B = {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a symplectic basis of (V, q). It can be shown that the class of ∑_{i=1}^{n} q(u_{i})q(v_{i}) in the quotient group F/℘(F), which is called the Arf invariant of (V, q) and is denoted by Δ(V, q) or Δ(q), is independent of the choice of B (see [14, Ch. 9, (4.2)]). In particular the Arf invariant of the form [a, b] is ab + ℘(F).

If (V, q) is a quadratic space, a nonzero vector v ∈ V is called isotropic if q(v) = 0 and anisotropic otherwise. A subspace W of V is called isotropic if there is an isotropic vector in W, otherwise W is called anisotropic. Every isotropic quadratic space is universal, see [12, Ch. I, (4.5)]. The subspace W is totally isotropic if q(w) = 0 for all w ∈ W. If W and W′ are two maximal totally isotropic subspaces of V, then one has dim W = dim W′ (see [4, (12.11)]). The dimension of a maximal totally isotropic subspace of V is called the Witt index of (V, q). By a quadratic plane we mean a 2-dimensional quadratic space (𝔼, φ). Every quadratic plane has a basis {u, v} such that B(u, v) = 1. For such a basis we have (uv)^{2} = uv + q(u)q(v) · 1, where this equality is considered in the Clifford algebra of (𝔼, φ). Two quadratic planes (𝔼_{1}, φ_{1}) and (𝔼_{2}, φ_{2}) are isometric as bilinear spaces but not necessarily isometric as quadratic spaces. It is known that every quadratic space is an orthogonal sum of quadratic planes (see [12, Ch. I, (4.3)]). Up to isometry, there is a unique isotropic quadratic plane (𝔼, φ), which is called a hyperbolic plane. For every hyperbolic plane (𝔼, φ) we have φ ≃ [1, 0]. We denote a hyperbolic plane by ℍ. A quadratic space (V, q) is called hyperbolic if it is an orthogonal sum of hyperbolic planes. Every hyperbolic space has trivial Arf invariant.

An isometry τ of (V, q) is called an involution in O(V, q), if τ^{2} = id. Let u ∈ V be an anisotropic vector and define a linear transformation τ_{u} : V → V via τ_{u}(v) = v + \frac{B(v,u)}{q(u)}u. It is easy to see that τ_{u} is an involution in O(V, q). Note that if B(u, v) = 0 then τ_{u}(v) = v, also τ_{u}(u) = u. We call τ_{u} the (orthogonal) reflection along u. In the literature, these isometries are also called orthogonal transvections. It is easy to see that if u ∈ V is an anisotropic vector, then the restriction of the inner automorphism Int(u) of C(V) to V is just the reflection τ_{u}. If (V, q) is a quadratic space, then by a theorem due to Dieudonné, every isometry σ of (V, q) is a product of reflections except for the case that dim V = 4, |F | = 2 and the Witt index of (V, q) is 2 (see [4, (14.16)]). If σ = τ_{u_{1}} ⋯ τ_{u_{k}} ∈ O(V, q), then it can be shown that the class of Π_{i=1}^{k}q(u_{i}) in the quotient group F^{×}/F^{×2}, which is called the spinor norm of σ and is denoted by θ(σ), is independent of the choice of the reflections τ_{u_{i}}’s, see [4, (14.17)]. In particular if σ = τ_{u} is a reflection, then the spinor norm of τ_{u} is the class of q(u) in the quotient group F^{×}/F^{×2}.

For an F -central simple algebra A the integer \sqrt{dim_{F} A} is called the degree of A. An involution on A is an anti-automorphism σ of A such that σ^{2} = id. Let (A, σ) be an F -central simple algebra with involution. Consider the subfield K of F consisting of the elements a of F such that σ(a) = a. It is known that either F = K, or F/K is a separable field extension of degree 2. In the first case we say σ is of the first kind and in the second case we say σ is of the second kind (or of unitary type).

Let V be a finite dimensional vector space over a field F and let B be a nondegenerate bilinear form over V. The unique involution σ_{B} on End_{F} (V), characterized by B(x, f(y)) = B(σ_{B}(f)(x), y) for every x, y ∈ V and f ∈ End_{F} (V), is called the adjoint involution of End_{F} (V) with respect to B.

Let (A, σ) be an F -central simple algebra of degree n with involution of the first kind. If L is a splitting field of A and V is an n-dimensional vector space over L, then there is a nondegenerate symmetric or antisymmetric bilinear form B over V such that (A_{L}, σ_{L}) ≃ (End_{L}(V), σ_{B}) (see [6, (2.1)]). An involution σ of the first kind is said to be of symplectic type if for any splitting field L of A and any isomorphism (A_{L}, σ_{L}) ≃ (End_{L}(V), σ_{B}), the bilinear form B is alternating. Otherwise we say that σ is of orthogonal type. For short, we say that the central simple algebra with involution (A, σ) is of orthogonal (resp. symplectic) type if σ is of orthogonal (resp. symplectic) type.

A quaternion algebra over a field F of characteristic 2, is a 4-dimensional F -algebra Q with a basis {1, i, j, k} subject to the relations i^{2} + i = α ∈ F, j^{2} = β ∈ F^{×}, ij = k and ij + ji = j. Note that these relations imply that k^{2} = (ij)^{2} = αβ. This algebra is denoted by [α, β) and we call {1, i, j, k} a standard basis of Q. The map γ : Q → Q defined by γ(a + bi + cj + dk) = a + b(i + 1) + cj + dk is an involution of the first kind on Q which is called the canonical involution of Q. The canonical involution γ is the unique involution on Q such that γ(x)x ∈ F for every x ∈ Q, see [14, Ch.8, (11.2)].

If (A, σ) is an F -central simple algebra with involution, the set of alternating elements in A is defined as follows:

Alt(A, σ) = {a − σ(a)|a ∈ A}.

If A is of even degree n = 2m over F and σ is of orthogonal type, the determinant of σ is the square class of the reduced norm of any alternating unit:

det σ = Nrd_{A}(a)F^{×2} ∈ F^{×}/F^{×2} for a ∈ Alt(A, σ) ∩ A^{∗},

and the discriminant of σ is the signed determinant:

disc σ = (−1)^{m} det σ ∈ F^{×}/F^{×2}.

Note that by [7, Lem. 2.1] or [6, (7.1)], the determinant and the discriminant are well-defined. Also if char F = 2 the discriminant and the determinant coincide.

3 Wiitala’s characterization of involutions of orthogonal groups in characteristic 2

Definition 3.1. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). We call the subspace Fix(V, τ) = {v ∈ V | τv = v}, the fixed subspace of τ.

We state here the following definition from [17].

Definition 3.2. Let (𝔸, q) be a quadratic space of dimension 4 over a field F of characteristic 2. An involution τ ∈ O(𝔸, q) is called an interchange isometry if the fixed subspace Fix(𝔸, τ) is a two dimensional totally isotropic space.

Remark 3.3. In [17], such maps are called “basic null involutions”. In this work have preferred to use the term interchange isometry. Also in view of (3.6) below, this naming is not unreasonable.

Lemma 3.4. Let (𝔸, q, B) be a quadratic space of dimension 4 over a field F of characteristic 2. If τ is an isometry of (𝔸, q), then τ is an interchange isometry if and only if there is a basis {w, x, y, z} of 𝔸 such that {x, w} is a basis of Fix(𝔸, τ), q(w) = q(x) = 0, τ(y) = x + y, τ(z) = w + z, B(x, z) = B(w, y) = 1 and all other pairs of vectors from this basis are orthogonal. In particular if τ is an interchange isometry of (𝔸, q), then (𝔸, q) is necessarily hyperbolic.

Proof. See [17, Lem. 2].

Remark 3.5. If (𝔸, q, B) is a 4-dimensional quadratic space over a field F of characteristic 2 and τ is an interchange isometry of (𝔸, q), the basis {w, x, y, z} of 𝔸 with the properties stated in (3.4), can be chosen with the additional property that q(y) = q(z) = 0; in fact set α = q(y), β = q(z), y′ = y + αw and z′ = z + βx. It is easy to see that the basis {w, x, y′, z′} has the desired property.

Proposition 3.6. Let (𝔸, q, B) be a quadratic space of dimension 4 over a field F of characteristic 2 and let τ be an isometry of (𝔸, q). The following statements are equivalent: (1) τ is an interchange isometry. (2) There exist hyperbolic planes ℍ_{1} and ℍ_{2} in 𝔸 such that 𝔸 = ℍ_{1} ⊥ ℍ_{2}, τ(ℍ_{1}) = ℍ_{2} and τ(ℍ_{2}) = ℍ_{1}.

(3) There exist quadratic planes 𝔼_{1} and 𝔼_{2} in 𝔸 such that 𝔸 = 𝔼_{1} ⊥ 𝔼_{2}, τ(𝔼_{1}) = 𝔼_{2} and τ(𝔼_{2}) = 𝔼_{1}.

Proof. (1) ⇒ (2): Let {w, x, y, z} be the basis of 𝔸 with the properties stated in (3.5) and set u_{1} = y, u_{2} = x + y = τ(u_{1}), v_{1} = w + z and v_{2} = z = τ(v_{1}). Then

B(u_{1}, u_{2}) = B(y, x + y) = 0,
B(u_{1}, v_{1}) = B(y, w + z) = B(y, w) + B(y, z) = 1 + 0 = 1,
B(u_{1}, v_{2}) = B(y, z) = 0,
B(v_{1}, u_{2}) = B(τ(v_{2}), τ(u_{1})) = B(v_{2}, u_{1}) = 0,
B(v_{1}, v_{2}) = B(w + z, z) = 0,
B(u_{2}, v_{2}) = B(τ(u_{1}), τ(v_{1})) = B(u_{1}, v_{1}) = 1.

As q(u_{1}) = q(y) = 0 and q(v_{2}) = q(z) = 0, the subspaces F u_{1} + F v_{1} and F u_{2} + F v_{2} of 𝔸 are hyperbolic. Let ℍ_{1} = F u_{1} + F v_{1} and ℍ_{2} = τ(ℍ_{1}) = F u_{2} + F v_{2}. Then ℍ_{1} = ℍ_{2}^{⊥}. Finally as τ^{2} = id, we obtain τ(ℍ_{2}) = ℍ_{1}.

(2) ⇒ (3) is immediate.

(3) ⇒ (1): Set u_{2} = τ(u_{1}) and v_{2} = τ(v_{1}). Then 𝔼_{2} = F u_{2} + F v_{2}. Let v = au_{1} + bu_{2} + cv_{1} + dv_{2} ∈ Fix(𝔸, τ), where a, b, c, d ∈ F. Then au_{2} + bu_{1} + cv_{2} + dv_{1} = au_{1} + bu_{2} + cv_{1} + dv_{2}. Hence a = b and c = d, i.e., v = a(u_{1} +u_{2})+c(v_{1} +v_{2}). On the other hand the subspace F (u_{1} +u_{2})+F (v_{1} +v_{2}) of 𝔸 is fixed by τ. So Fix(𝔸, τ) = F (u_{1} + u_{2}) + F (v_{1} + v_{2}). Finally we have q(u_{1}+u_{2}) = q(v_{1}+v_{2}) = B(u_{1}+u_{2}, v_{1}+v_{2}) = 0, i.e., Fix(𝔸, τ) is a 2-dimensional totally isotropic space. So τ is an interchange isometry.

Theorem 3.7. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an isometry of (V, q). Then τ is an involution in O(V, q) if and only if there is a decomposition of V into regular subspaces V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} such that:

(a) r and s are nonnegative integers and for every i, dim 𝔼_{i} = 2 and dim 𝔸_{i} = 4.

(b) τ maps each subspace W, 𝔼_{i} and 𝔸_{i} into itself.

(c) τ|_{W} = id.

(d) τ|_{𝔼_{i}} is a reflection.

(e) τ|_{𝔸_{i}} is an interchange isometry.

Proof. See [17, Thm. 1].

Definition 3.8. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). We call the decomposition V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} in (3.7), a Wiitala decomposition of (V, τ) and we call the subspace W, a maximal fixed orthogonal summand of (V, τ); in view of (3.9) below, this naming is justified. For every Wiitala decomposition V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} of (V, τ), we have a decomposition τ = id_{W} ⊥ τ_{1} ⊥ ⋯ ⊥ τ_{r} ⊥ τ′_{1} ⊥ ⋯ ⊥ τ′_{s} which is called a Wiitala decomposition of τ where τ_{i} is a reflection of 𝔼_{i} for i = 1, ⋯, r and τ′_{i} is an interchange isometry of 𝔸_{i} for i = 1, ⋯, s (i.e., the integers r and s are the respective number of reflections and interchange isometries in the above Wiitala decomposition of τ).

Proposition 3.9. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Let U ⊆ V be an orthogonal summand of V such that τ|_{U} = id. Then there exists a maximal fixed orthogonal summand W of (V, τ) such that U ⊆ W. In particular dim W is uniquely determined by τ.

Proof. Let U′ be an orthogonal complement of U, i.e., V = U ⊥ U′. Then τ|_{U′} is an involution in O(U′, q|_{U′}). Let U′ = W_{1} ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (U′, τ|_{U′}) and set W = U ⊥ W_{1}. Then V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} is a Wiitala decomposition of (V, τ). So W is a maximal fixed orthogonal summand of (V, τ) and U ⊆ W.

Remark 3.10. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Let W be a maximal fixed orthogonal summand of (V, τ). Then W is not necessarily unique up to isometry. For example let (𝔼_{i}, φ_{i}) ≃ [α, β], i = 1, 2, be an anisotropic quadratic plane over F with standard symplectic basis {u_{i}, v_{i}}. Set (V, q) = (𝔼_{1}, φ_{1}) ⊥ (𝔼_{2}, φ_{2}) and τ = id_{𝔼_{1}} ⊥ τ_{u_{2}} where τ_{u_{2}} is the reflection of 𝔼_{2} along u_{2}. Then τ is an involution in O(V, q) and W_{1} = 𝔼_{1} is a maximal fixed orthogonal summand of (V, τ). Now set W = F (u_{1} + u_{2}) + F v_{1} and 𝔼 = F u_{2} + F (v_{1} + v_{2}). Then V = W ⊥ 𝔼 and τ|_{W} = id. Also τ|_{𝔼} is the reflection along u_{2}, because τ(u_{2}) = u_{2} = τ_{u_{2}}(u_{2}) and τ(v_{1} + v_{2}) = v_{1} + (v_{2} + \frac{1}{q(u_{2})}u_{2}) = (v_{1} + v_{2}) + \frac{B(v_{1}+v_{2},u_{2})}{q(u_{2})}u_{2} = τ_{u_{2}}(v_{2}). So V = W ⊥ 𝔼 is a Wiitala decomposition of (V, τ) and W is a maximal fixed orthogonal summand of (V, τ). Now we have q(u_{1} + u_{2}) = α + α = 0, so W is isotropic and W_{1} ≄ W.

Definition 3.11. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Let V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (V, τ). We call a symplectic basis

{u_{1}, v_{1}, ⋯, u_{t}, v_{t}, u′_{1}, v′_{1}, ⋯, u′_{r}, v′_{r}, u′′_{1}, v′′_{1}, ⋯, u′′_{2s}, v′′_{2s}}

of V a τ-symplectic basis of (V, τ) if it satisfies the following conditions:

(a) The set {u_{1}, v_{1}, ⋯, u_{t}, v_{t}} is a symplectic basis of the subspace W (so t = \frac{1}{2} dim W).

(b) For 1 ⩽ i ⩽ r, {u′_{i}, v′_{i}} is a symplectic basis of the subspace 𝔼_{i} such that q(u′_{i}) ≠ 0, τ(u′_{i}) = u′_{i} and τ(v′_{i}) = v′_{i} + \frac{1}{q(u′_{i})}u′_{i} (i.e., τ|_{𝔼_{i}} = τ_{u′_{i}}).

(c) For 1 ⩽ i ⩽ s, {u′′_{2i−1}, v′′_{2i−1}, u′′_{2i}, v′′_{2i}} is a basis of the subspace 𝔸_{i} such that the subspace F u′′_{2i−1}+F v′′_{2i−1} is perpendicular to the subspace F u′′_{2i}+F v′′_{2i}, τ(u′′_{2i−1}) = u′′_{2i}, τ(u′′_{2i}) = u′′_{2i−1}, τ(v′′_{2i−1}) = v′′_{2i} and τ(v′′_{2i}) = v′′_{2i−1} (i.e., τ|_{𝔸_{i}} is an interchange isometry).

For simplicity, we denote the above τ-symplectic basis of V by {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} where n = t + r + 2s and for t + 1 ⩽ i ⩽ t + r, we have u_{i} = u′_{i−t} and v_{i} = v′_{i−t} and for t + r + 1 ⩽ i ⩽ t + r + 2s = n, we have u_{i} = u′′_{i−t−r} and v_{i} = v′′_{i−t−r}.

Remark 3.12. Let (𝔼, φ) be a quadratic plane over a field F of characteristic 2 and let τ ≠ id be an involution in O(𝔼, φ). Then τ is a reflection (see [17, Lem. 1]). In particular for every quadratic plane (𝔼, φ) and every involution τ ∈ O(𝔼, φ), there exists an anisotropic vector u ∈ 𝔼 such that τ(u) = u.

Lemma 3.13. Let (𝔼, φ) be a quadratic plane over a field F of characteristic 2 and let τ be an involution in O(𝔼, φ). Let t = dim Fix(𝔼, τ) − \frac{1}{2} dim 𝔼 and let u ∈ 𝔼 be an anisotropic vector such that τ(u) = u. Then for every x ∈ 𝔼, we have τ(x) = x + (t + 1)B(x, u)u^{−1}.

Proof. If τ = id, then t = 1 and τ(x) = x = x + (t + 1)B(x, u)u^{−1}. Otherwise by (3.12), we have τ = τ_{u}, t = 0 and τ(x) = x + B(x, u)u^{−1} = x + (t + 1)B(x, u)u^{−1}.

Lemma 3.14. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q).

(a) If dim V = 2 then we have τ = id if and only if for every v ∈ V we have B(v, τ(v)) = 0.

(b) If dim V = 4 and τ ≠ id, then τ is an interchange isometry if and only if for every v ∈ V, we have B(v, τ(v)) = 0.

(c) We have B(v, τ(v)) = 0 for every v ∈ V if and only if there is no reflection in any Wiitala decomposition of τ.

Proof. (a) If τ = id, the conclusion is evident. Conversely suppose that τ ≠ id. By (3.12) there exists an anisotropic vector u ∈ V such that τ is the reflection along u, i.e., τ = τ_{u}. Let v ∈ V with B(u, v) = 1. Then B(v, τ(v)) = B(v, v + \frac{1}{q(u)}u) = \frac{1}{q(u)} ≠ 0.

(b) Let τ be an interchange isometry. Choose a basis {w, x, y, z} of V with the properties stated in (3.4). Let v ∈ V and write v = aw + bx + cy + dz where a, b, c, d ∈ F. We have

B(v, τ(v)) = B(aw + bx + cy + dz, aw + bx + c(y + x) + d(z + w))
= B(aw + bx + cy + dz, (a + d)w + (b + c)x + cy + dz)
= ac + bd + c(a + d) + d(b + c) = 0,

and we are done. Conversely suppose that B(v, τ(v)) = 0 for every v ∈ V. As τ ≠ id, if τ is not an interchange isometry then by (3.7) there is necessarily a quadratic plane 𝔼 ⊆ V such that τ|_{𝔼} is a reflection. But (a) implies that there exists v ∈ 𝔼 such that B(v, τ(v)) ≠ 0, this contradicts the hypothesis.

(c) If there is a reflection in a Wiitala decomposition of τ, then there exists v ∈ V such that B(v, τ(v)) ≠ 0. Conversely suppose that there is no reflection in any Wiitala decomposition of τ. By (3.7), we can write V = W ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s}, where τ|_{W} = id and τ|_{𝔸_{i}} is an interchange isometry. For every v ∈ V we can write v = v_{0} + v_{1} + ⋯ + v_{s} where v_{0} ∈ W and v_{i} ∈ 𝔸_{i}, i = 1, …, s. Then using (b) we obtain

B(v, τ(v)) = B(v_{0}, τ(v_{0})) + B(v_{1}, τ(v_{1})) + ⋯ + B(v_{s}, τ(v_{s}))
= 0 + 0 + ⋯ + 0 = 0.

Proposition 3.15. Let (V, q, B) be a quadratic space of dimension 6 over a field F of characteristic 2 and let τ be an involution in O(V, q). Suppose that V has a decomposition of the form V = 𝔼 ⊥ 𝔸 where 𝔼 is a quadratic plane in V, τ|_{𝔼} is a reflection and τ|_{𝔸} is an interchange isometry. Then there is a decomposition of V of the form V = 𝔼_{1} ⊥ 𝔼_{2} ⊥ 𝔼_{3} such that 𝔼_{1}, 𝔼_{2} and 𝔼_{3} are quadratic planes in V and for every i = 1, 2, 3, τ|_{𝔼_{i}} is a reflection.

Proof. See [17, Thm. 2].

Theorem 3.16. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Then there is a decomposition of (V, τ) in exactly one of the following forms:

(a) V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} where τ|_{W} = id, 𝔼_{i} is a quadratic subplane of V and τ|_{𝔼_{i}} is a reflection for every 1 ⩽ i ⩽ r.

(b) V = W ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} where τ|_{W} = id, dim 𝔸_{i} = 4 and τ|_{𝔸_{i}} is an interchange isometry for every 1 ⩽ i ⩽ s.

Furthermore the type of decomposition occurring and the numbers r and s are uniquely determined by τ.

Proof. See [17, Thm. 2].

Definition 3.17. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). We say that τ is of reflectional (resp. interchanging) kind if V has a decomposition of the form (a) (resp. (b)) in (3.16).

The following result is a restatement of (3.14 (c)):

Corollary 3.18. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Then τ is of interchanging kind if and only if B(v, τ(v)) = 0 for every v ∈ V.

Proposition 3.19. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Let r be the number of reflections in any Wiitala decomposition of (V, τ). Then r ≡ dim Fix(V, τ) mod 2 and dim Fix(V, τ) = \frac{1}{2} dim W + \frac{1}{2} dim V where W is a maximal fixed orthogonal summand of (V, τ). In particular dim Fix(V, τ) ⩾ \frac{1}{2} dim V. Also we have dim Fix(V, τ) = \frac{1}{2} dim V if and only if W = 0.

Proof. Let V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (V, τ). Then for 1 ⩽ i ⩽ r, τ|_{𝔼_{i}} is a reflection, so the fixed subspace of τ|_{𝔼_{i}} is one dimensional. Similarly for 1 ⩽ i ⩽ s, τ|_{𝔸_{i}} is an interchange isometry, so the fixed subspace of τ|_{𝔸_{i}} is two dimensional. So the fixed subspace of the restriction of τ to 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} has dimension r + 2s and we obtain dim Fix(V, τ) = r + 2s + dim W = (r + 2s + \frac{1}{2} dim W) + \frac{1}{2} dim W = \frac{1}{2} dim V + \frac{1}{2} dim W. Finally since W is a regular subspace of v, dim W is even, so r ≡ dim Fix(V, τ) mod 2.

Remark 3.20. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). If dim Fix(V, τ) = \frac{1}{2} dim V and τ is of interchanging kind, then by (3.19), (V, τ) has trivial maximal fixed orthogonal summand. So by (3.16), V = 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s}. By (3.6), V is a hyperbolic space. In particular the Arf invariant of q is trivial. If τ is of reflectional kind, depending on (V, q), Δ(q) can be trivial or nontrivial. For example if V = ℍ and τ is a reflection of V, then Δ(q) is trivial, while if V = 𝔼 is the quadratic plane [1, δ] with δ ∈ F \℘(F) and τ is a reflection of V, then Δ(q) = δ+℘(F) ∈ F/℘(F) is nontrivial.

4 Generalities on some natural involutions of a Clifford algebra

Remark 4.1. Let (V, q) be a quadratic space over a field F of characteristic 2 and let C(V) be the Clifford algebra of (V, q). If τ : V → V is an involution in O(V, q), then τ induces an involution J_{τ}^{q} on C(V) such that J_{τ}^{q}(v) = τ(v) for every v ∈ V. Also if σ is an involution on C(V) such that σ(V) = V, then τ = σ|_{V} is involution in O(V, q). We call J_{τ}^{q} the involution induced by τ on C(V) and we usually abbreviate J_{τ}^{q} by J_{τ}.

Remark 4.2. Let (𝔼, φ) be a quadratic plane over a field F of characteristic 2 and let τ = τ_{u} be a reflection along some anisotropic vector u ∈ 𝔼. Extend {u} to a symplectic basis {u, v} of 𝔼. Consider the involution J_{τ} on C(𝔼). We have J_{τ} (u) = u, J_{τ} (v) = v + \frac{1}{q(u)}u and J_{τ} (uv) = τ(v)τ(u) = (v + \frac{1}{q(u)}u)u = vu + 1 = uv. So if x = a + bu + cv + duv ∈ C(𝔼) where a, b, c, d ∈ F, then we get x−J_{τ} (x) = \frac{c}{q(u)}u, i.e., Alt(C(𝔼), J_{τ}) = F u. In particular we have x−J_{τ} (x) ∈ F if and only if x = J_{τ} (x).

Proposition 4.3. Let (𝔼, φ, B) be a quadratic plane over a field F of characteristic 2 and let τ be an involution in O(𝔼, φ). The involution J_{τ} on C(𝔼) is of symplectic type if and only if τ = id if and only if D(τ) = 0 ∈ ℤ/2ℤ. In other words J_{τ} is of orthogonal type if and only if τ is a reflection.

Proof. Suppose that τ = id and {x, y} is a basis of 𝔼 such that B(x, y) = 1. Then J_{τ} (xy) = τ(y)τ(x) = yx = xy + 1, so 1 ∈ Alt(C(𝔼), J_{τ}) and by [6, (2.6)], J_{τ} is of symplectic type.

Conversely suppose that τ ≠ id. Then by (3.12), τ is a reflection along some anisotropic vector u ∈ 𝔼, i.e., τ = τ_{u}. By (4.2), Alt(C(𝔼), J_{τ}) = F u, hence 1 ∉ Alt(C(𝔼), J_{τ}). So by [6, (2.6)], J_{τ} is of orthogonal type.

This proves the first part of the proposition. For the second part, if τ ≠ id is an involution in O(𝔼, φ), then τ is a reflection and for any reflection τ we have D(τ) = 1 ∈ ℤ/2ℤ (see [14, Ch. 9, (4.11)]).

Lemma 4.4. Let (V_{1}, q_{1}) and (V_{2}, q_{2}) be quadratic spaces over a field F of characteristic 2. If τ_{1} and τ_{2} are involutions in O(V_{1}, q_{1}) and O(V_{2}, q_{2}), respectively, then (C(V_{1}), J_{τ_{1}}) ⊗ (C(V_{2}), J_{τ_{2}}) ≃ (C(V_{1} ⊥ V_{2}), J_{τ_{1}⊥τ_{2}}).

Proof. It is easy to see that the linear map φ : V_{1} ⊥ V_{2} → C(V_{1})⊗C(V_{2}) defined by φ(v_{1} + v_{2}) = v_{1} ⊗ 1 + 1 ⊗ v_{2} where v_{1} ∈ V_{1} and v_{2} ∈ V_{2} can be extended to an isomorphism φ : C(V_{1} ⊥ V_{2}) ≃ C(V_{1}) ⊗ C(V_{2}) (cf. [4, p. 120]). So it remains to show that φ ∘ J_{τ_{1}⊥τ_{2}} = (J_{τ_{1}} ⊗ J_{τ_{2}}) ∘ φ. It is enough to check that φ ∘ J_{τ_{1}⊥τ_{2}}(w) = (J_{τ_{1}} ⊗ J_{τ_{2}}) ∘ φ(w) for every w ∈ V_{1} ⊥ V_{2}. Write w = v_{1} + v_{2} where v_{1} ∈ V_{1} and v_{2} ∈ V_{2}, then

φ ∘ J_{τ_{1}⊥τ_{2}}(w) = φ ∘ J_{τ_{1}⊥τ_{2}}(v_{1} + v_{2}) = φ(τ_{1}(v_{1}) + τ_{2}(v_{2}))
= τ_{1}(v_{1}) ⊗ 1 + 1 ⊗ τ_{2}(v_{2}) = (J_{τ_{1}} ⊗ J_{τ_{2}})(v_{1} ⊗ 1 + 1 ⊗ v_{2})
= (J_{τ_{1}} ⊗ J_{τ_{2}}) ∘ φ(v_{1} + v_{2}) = (J_{τ_{1}} ⊗ J_{τ_{2}}) ∘ φ(w).

Lemma 4.5. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Then for every a ∈ F^{×} we have (C_{0}(V, q), J_{τ}^{q}) ≅ (C_{0}(V, a · q), J_{τ}^{a·q}).

Proof. Define a bilinear map ψ : V × V → C_{0}(V, a · q) via ψ(x, y) = a^{−1} · xy for every x, y ∈ V. For every x, y, z ∈ V we have ψ(x, y)ψ(y, z) = (a^{−1} · xy)(a^{−1} · yz) = a^{−2}aq(y) · xz = a^{−1}q(y) · xz = q(y) · ψ(x, z) and ψ(x, x) = a^{−1} · x^{2} = a^{−1}a · q(x) = q(x) · 1. So ψ is an even Clifford map and by (A.5), there exists an F -algebra homomorphism Ψ : C_{0}(V, q) → C_{0}(V, a · q) induced by Ψ(xy) = a^{−1} · xy for every x, y ∈ V. The map ψ is clearly an isomorphism.

Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a symplectic basis of (V, q). For every 1 ⩽ i, j ⩽ n we have

Ψ ∘ J_{τ}^{q}(u_{i}u_{j})=Ψ(τ(u_{j})τ(u_{i}))=a^{−1}τ(u_{j})τ(u_{i})=J_{τ}^{a·q}(a^{−1}u_{i}u_{j})=J_{τ}^{a·q} ∘ Ψ(u_{i}u_{j}).

Similarly we have the following relations

Ψ ∘ J_{τ}^{q}(v_{i}v_{j}) = J_{τ}^{a·q} ∘ Ψ(v_{i}v_{j}),
Ψ ∘ J_{τ}^{q}(u_{i}v_{j}) = J_{τ}^{a·q} ∘ Ψ(u_{i}v_{j}).

So Ψ∘J_{τ}^{q}(x) = J_{τ}^{a·q} ∘Ψ(x) for every x ∈ C_{0}(V, q). This completes the proof.

Proposition 4.6. Let (𝔸, q) be a quadratic space of dimension 4 over a field F of characteristic 2 and let τ be an interchange isometry of (𝔸, q). Then the involution J_{τ} on C(𝔸) is of orthogonal type.

Proof. Let (𝔼, φ) be a quadratic plane over F and let ρ be a reflection of (𝔼, φ). Set V = 𝔼 ⊥ 𝔸. Then τ ⊥ ρ is an involution in O(V, φ ⊥ q). By (3.15), there is a decomposition V = 𝔼_{1} ⊥ 𝔼_{2} ⊥ 𝔼_{3} such that τ_{i} := (τ ⊥ ρ)|_{𝔼_{i}} is a reflection for every i = 1, 2, 3. Hence by (4.3), J_{τ_{i}} is an involution of orthogonal type on C(𝔼_{i}). Using (4.4) we obtain (C(V), J_{τ⊥ρ}) ≃ (C(𝔼_{1}), J_{τ_{1}})⊗(C(𝔼_{2}), J_{τ_{2}})⊗(C(𝔼_{3}), J_{τ_{3}}), and by [6, (2.23)], J_{τ⊥ρ} is of orthogonal type.

On the other hand we have (C(V), J_{τ⊥ρ}) ≃ (C(𝔸), J_{τ}) ⊗ (C(𝔼), J_{ρ}). Since (C(V), J_{τ⊥ρ}) is of orthogonal type, again by [6, (2.23)], J_{τ} is of orthogonal type on C(𝔸) and we are done.

Theorem 4.7. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Then the involution J_{τ} on C(V) is of orthogonal type if and only if (V, τ) has trivial maximal fixed orthogonal summand if and only if dim Fix(V, τ) = \frac{1}{2} dim V.

Proof. Let V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (V, τ). We have (C(V), J_{τ}) ≃ (C(W), J_{id}) ⊗ (C(W′), J_{τ|_{W′}}) where W′ = 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s}. The conclusion follows from (4.3), (4.4), (4.6) and [6, (2.23)]. The equivalence of the last two statements follows from (3.19).

Corollary 4.8. Let (V, q) be a quadratic space over a field F of characteristic 2.

(a) The involution J_{id} on C(V) is of symplectic type.

(b) Let τ be a reflection of V. If dim V = 2, then the involution J_{τ} on C(V) is of orthogonal type. If dim V ⩾ 4, then J_{τ} is of symplectic type.

Proof. (a) We have dim Fix(V, id) = dim V > \frac{1}{2} dim V. So by (4.7), J_{id} is of symplectic type.

(b) If dim V = 2, then by (4.3), J_{τ} is of orthogonal type. If dim V ⩾ 4, then dim Fix(V, τ) = dim V − 1 > \frac{1}{2} dim V and by (4.7), J_{τ} is of symplectic type.

Proposition 4.9. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Then D(τ) is equal to the class of dim Fix(V, τ) in ℤ/2ℤ.

Proof. Let V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (V, τ) and let {u_{1}, v_{1}, ⋯, u_{t}, v_{t}, u′_{1}, v′_{1}, ⋯, u′_{r}, v′_{r}, u′′_{1}, v′′_{1}, ⋯, u′′_{2s}, v′′_{2s}} be a τ-symplectic basis with respect to this decomposition and set z = u_{1}v_{1} + ⋯ + u_{t}v_{t} + u′_{1}v′_{1} + ⋯ + u′_{r}v′_{r} + u′′_{1}v′′_{1} + ⋯ + u′′_{2s}v′′_{2s}. Let I_{τ} be the automorphism of C_{0}(V) induced by τ. Then I_{τ} |_{W} = id. Also for 1 ⩽ i ⩽ r, we have I_{τ} (u′_{i}v′_{i}) = u′_{i}(v′_{i} + \frac{1}{q(u′_{i})}u′_{i}) = u′_{i}v′_{i} + 1. Finally for 1 ⩽ i ⩽ s, we have I_{τ} (u′′_{2i−1}v′′_{2i−1} + u′′_{2i}v′′_{2i}) = u′′_{2i}v′′_{2i} + u′′_{2i−1}v′′_{2i−1}. Hence I_{τ} (z) = z + r · 1, where r is the number of reflections in the above Wiitala decomposition of τ. Note that by (3.19) the number of reflections modulo two is independent of the decomposition of τ. Since Z(C_{0}(V)) is generated by 1 and z over F (see [4, (13.12)]), the Dickson invariant D(τ) is equal to the class of r in ℤ/2ℤ. On the other hand by (3.19), r ≡ dim Fix(V, τ) mod 2 and therefore, D(τ) is equal to the class of dim Fix(V, τ) in ℤ/2ℤ.

Lemma 4.10. Let F be a field of characteristic 2 and let A a be central simple algebra over F. If [A : F] = n^{2} is even and x ∈ A is an element such that x^{2} = α ∈ F, then Nrd(x) = α^{\frac{n}{2}} .

Proof. We have Nrd(x)^{2} = Nrd(α) = α^{n}. Since n is even and char F = 2, we get Nrd(x) = α^{\frac{n}{2}} .

Proposition 4.11. Let F be a field of characteristic 2 and let (V, q) be a quadratic space over F. Let τ be an involution in O(V, q) such that J_{τ} is of orthogonal type (in other words dim Fix(V, τ) = \frac{1}{2} dim V) on C(V).

(a) If dim V = 2, then disc J_{τ} = θ(τ).

(b) If dim V ⩾ 4, then disc J_{τ} = 1 ∈ F^{×}/F^{×2}.

Proof. (a) Since J_{τ} is of orthogonal type, by (4.3), we have τ = τ_{u} where u ∈ V is an anisotropic vector. By (4.2), we have u ∈ Alt(C(V), J_{τ_{u}}). On the other hand since u^{2} = q(u) ≠ 0, the vector u is invertible in C(V) and by (4.10), Nrd(u) = q(u). So disc J_{τ_{u}} = Nrd(u)F^{×2} = q(u)F^{×2} = θ(τ_{u}).

(b) By (4.7), (V, τ) has trivial maximal fixed orthogonal summand, so (V, τ) has a Wiitala decomposition V = 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s}. If dim V = 4, then either τ = τ_{1} ⊥ τ_{2} where τ_{i}, i = 1, 2, is a reflection of 𝔼_{i} or τ is an interchange isometry of V. In the first case we have (C(V), J_{τ}) ≃ (C(𝔼_{1}), J_{τ_{1}})⊗ (C(𝔼_{2}), J_{τ_{2}}). Using [6, (7.3)], we have disc J_{τ} = disc(J_{τ_{1}} ⊗ J_{τ_{2}}) = 1. Now consider the case where τ is an interchange isometry. Let {w, x, y, z} be the basis of V stated in (3.4). Then J_{τ} (xz) = τ(z)τ(x) = (w + z)x = wx + zx = wx + xz + 1, i.e., wx + 1 ∈ Alt(C(V), J_{τ}). Since B(x, w) = 0 we have wx = xw. We therefore obtain (wx + 1)^{2} = wxwx + 1 = w^{2}x^{2} + 1 = q(w)q(x) + 1 = 1. So wx+1 ∈ C(V)^{∗} and by (4.10), Nrd(wx+1) = 1. So disc J_{τ} = Nrd(wx+1)F^{×2} = 1 ∈ F^{×}/F^{×2}.

If dim V > 4, then using the Wiitala decomposition of (V, τ), there exist two nonzero subspaces V_{1} and V_{2} of V such that V = V_{1} ⊥ V_{2} and V_{1} and V_{2} are stable under τ. So (C(V), J_{τ}) ≃ (C(V_{1}), J_{τ_{1}}) ⊗ (C(V_{2}), J_{τ_{2}}), where τ_{i} = τ|_{V_{i}}, i = 1, 2, and by [6, (7.3)], disc J_{τ} = disc(J_{τ_{1}} ⊗ J_{τ_{2}}) = 1.

Lemma 4.12. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Let {u_{1}, v_{1}, ⋯, u_{t}, v_{t}, u′_{1}, v′_{1}, ⋯, u′_{r}, v′_{r}, u′′_{1}, v′′_{1}, ⋯, u′′_{2s}, v′′_{2s}} be a τ-symplectic basis of V. Set z = u_{1}v_{1} + ⋯ + u_{t}v_{t} + u′_{1}v′_{1} + ⋯ + u′_{r}v′_{r} + u′′_{1}v′′_{1} + ⋯ + u′′_{2s}v′′_{2s} ∈ C_{0}(V) and δ = q(u_{1})q(v_{1}) + ⋯ + q(u_{t})q(v_{t}) + q(u′_{1})q(v′_{1}) + ⋯ + q(u′_{r})q(v′_{r}) + q(u′′_{1})q(v′′_{1}) + ⋯ + q(u′′_{2s})q(v′′_{2s}) ∈ F. Then z^{2} = z + δ · 1 and J_{τ} (z) = z + t · 1. (Note that according to (3.19), t = dim Fix(V, τ) − \frac{1}{2} dim V).

Proof. By [4, p. 123], we have z^{2} = z+δ·1. Let V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (V, τ) such that {u_{1}, v_{1}, ⋯, u_{t}, v_{t}, u′_{1}, v′_{1} , ⋯, u′_{r}, v′_{r}, u′′_{1}, v′′_{1}, ⋯, u′′_{2s}, v′′_{2s}} is a τ-symplectic basis with respect to this decomposition and t = dim Fix(V, τ) − \frac{1}{2} dim V. For 1 ⩽ i ⩽ t, we have J_{τ} (u_{i}v_{i}) = τ(v_{i})τ(u_{i}) = v_{i}u_{i} = u_{i}v_{i} + 1. Similarly for 1 ⩽ i ⩽ r, we have J_{τ} (u′_{i}v′_{i}) = τ(v′_{i})τ(u′_{i}) = (v′_{i} + \frac{1}{q(u′_{i})}u′_{i})u′_{i} = v′_{i}u′_{i} +1 = u′_{i}v′_{i}. Finally for 1 ⩽ i ⩽ s, we have J_{τ} (u′′_{2i−1}v′′_{2i−1} + u′′_{2i}v′′_{2i}) = v′′_{2i}u′′_{2i} + v′′_{2i−1}u′′_{2i−1} = u′′_{2i−1}v′′_{2i−1} + u′′_{2i}v′′_{2i}. We therefore obtain J_{τ} (z) = z + t · 1 and the result is proved.

Theorem 4.13. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Let t = dim Fix(V, τ) − \frac{1}{2} dim V. Suppose that Δ(q) is nontrivial. Consider the involution J_{τ} on C_{0}(V):

(a) If t ≡ 0 mod 2 then J_{τ} is of the first kind. Furthermore if t = 0 then J_{τ} is of orthogonal type and if t ≠ 0 then J_{τ} is of symplectic type.

(b) If t ≡ 1 mod 2 then J_{τ} is of the second kind.

Proof. Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a τ-symplectic basis of V and let W be a maximal fixed orthogonal summand of (V, τ). Set z = u_{1}v_{1} + ⋯ + u_{n}v_{n}. Then by (4.12), we have J_{τ} (z) = t·1+z where t = \frac{1}{2} dim W = dim Fix(V, τ)− \frac{1}{2} dim V. By [4, (13.12)], Z(C_{0}(V)) is generated as an F -algebra by 1 and z over F. If t ≡ 0 mod 2 then J_{τ} (z) = z, i.e., J_{τ} |_{Z(C_{0}(V))} = id. Hence J_{τ} is of the first kind. Now if t = 0, then W = 0 and by (4.7), J_{τ} is of orthogonal type on C(V), so by [6, (2.6)] 1 ∉ Alt(C(V), J_{τ}). So 1 ∉ Alt(C_{0}(V), J_{τ}) and by [6, (2.6)], J_{τ} is of orthogonal type on C_{0}(V). Similarly if t ≠ 0, then t ⩾ 2, so u_{1}, v_{1} ∈ W and J_{τ} (u_{1}v_{1}) = v_{1}u_{1} = u_{1}v_{1} + 1. Hence 1 ∈ Alt(C_{0}(V), J_{τ}) and again by [6, (2.6)], J_{τ} is of symplectic type.

If t ≡ 1 mod 2 then J_{τ} (z) = z + 1, so J_{τ} is of the second kind.

Remark 4.14. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). If Δ(q) is nontrivial, then C_{0}(V) is a central simple algebra over Z = Z(C_{0}(V)) and Z/F is a quadratic field extension. More precisely if {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} is a symplectic basis of V and z = u_{1}v_{1} + ⋯ + u_{n}v_{n}, then Z = F (z). If x = α + βz ∈ Z for α, β ∈ F and x^{2} ∈ F, then by (4.12) we have α^{2} + β^{2}(z + δ) ∈ F where δ = ∑_{i=1}^{n} q(u_{i})q(v_{i}). So β = 0 and x ∈ F. This shows that Z^{×2} ∩ F^{×} = F^{×2}. Therefore, there exists a natural injection F^{×}/F^{×2} ↪ Z^{×}/Z^{×2} and we may consider F^{×}/F^{×2} as a subgroup of Z^{×}/Z^{×2}.

Proposition 4.15. Let (V, q) be a quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Suppose that Δ(q) is nontrivial and J_{τ} is an involution of orthogonal type (in other words dim Fix(V, τ) = \frac{1}{2} dim V) on C_{0}(V). Let i : F^{×}/F^{×2} ↪ Z^{×}/Z^{×2} be the natural injection discussed in (4.14).

(a) If dim V = 4, then disc J_{τ} = i(θ(τ)).

(b) If dim V ⩾ 6, then disc J_{τ} is trivial.

Proof. Since Δ(q) is nontrivial, by [4, (13.12)], C_{0}(V) is a central simple algebra. Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a τ-symplectic basis of V. By (3.19), (V, τ) has trivial maximal fixed orthogonal summand, so (V, τ) has a Wiitala decomposition V = 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s}. If τ is of interchanging kind, then by (3.20), Δ(q) is trivial which contradicts the assumption. So τ is of reflectional kind and we may assume that τ is an orthogonal sum of reflections τ_{i} = τ_{u_{i}} of 𝔼_{i}. We have J_{τ} (u_{1}v_{2}) = τ_{2}(v_{2})τ_{1}(u_{1}) = (v_{2} + \frac{1}{q(u_{2})}u_{2})u_{1} = v_{2}u_{1} + \frac{1}{q(u_{2})}u_{2}u_{1} = u_{1}v_{2} + \frac{1}{q(u_{2})}u_{1}u_{2}, i.e., u_{1}u_{2} ∈ Alt(C_{0}(V), J_{τ}). We also have (u_{1}u_{2})^{2} = q(u_{1})q(u_{2}) ≠ 0, so u_{1}u_{2} ∈ C_{0}(V)^{∗}.

(a) By (4.10), Nrd(u_{1}u_{2}) = q(u_{1})q(u_{2}). Therefore, disc J_{τ} = Nrd(u_{1}u_{2})Z^{×2} = q(u_{1})q(u_{2})Z^{×2}. Finally note that we have τ = τ_{u_{1}}τ_{u_{2}} where we consider τ_{u_{i}} as a reflection of V and therefore, disc J_{τ} = i(θ(τ)) = q(u_{1})q(u_{2})Z^{×2}.

(b) As dim V ⩾ 6, we obtain deg(C_{0}(V)) ≡ 0 mod 4. Since (u_{1}u_{2})^{2} = q(u_{1})q(u_{2}), by (4.10), we have Nrd(u_{1}u_{2}) = 1, so disc J_{τ} = Nrd(u_{1}u_{2})Z^{×2} is trivial.

Remark 4.16. The assumption that Δ(q) is nontrivial in (4.15) is necessary, because by [4, (13.12)], if Δ(q) is trivial, then C_{0}(V) is not central simple algebra and the reduced norm is not defined. Also we have assumed that dim V ⩾ 4, because if dim V = 2 then J_{τ} is identity and disc J_{τ} is not defined.

5 On a result of Mammone, Tignol and Wadsworth

In [10] it has been shown that if (𝔼, φ) is the quadratic plane [a, b] where a ≠ 0, then for any quadratic space (V, q) we have C(𝔼 ⊥ V, φ ⊥ q) ≃ C(𝔼′, φ′) ⊗ C(V, a · q) and C_{0}(𝔼 ⊥ V, φ ⊥ q) ≃ C_{0}(𝔼′′, φ′′) ⊗ C(V, a · q), where (𝔼′, φ′) and (𝔼′′, φ′′) are quadratic planes [a, b + a^{−1}δ_{1}] and [1, δ_{1} + δ_{2}], respectively, here δ_{1} ∈ F is a representative of the class Δ(q) ∈ F/℘(F) and δ_{2} ∈ F is a representative of the class Δ(φ) ∈ F/℘(F), i.e., Δ(q) = δ_{1} + ℘(F) ∈ F/℘(F) and Δ(φ) = δ_{2} + ℘(F) ∈ F/℘(F). We complement these results by showing that:

Proposition 5.1. Let F be a field of characteristic 2 and let (𝔼, φ) and (V, q) be two quadratic spaces over F such that dim 𝔼 = 2.

(i) If φ ≃ [a, b] for a, b ∈ F and a ≠ 0, then C(𝔼 ⊥ V, φ ⊥ q) ≃ C(𝔼′, φ′) ⊗ C(V, a·q) ≃ C(𝔼′ ⊥ V, φ′ ⊥ a·q) where (𝔼′, φ′) is the quadratic plane [a, b+a^{−1}δ] and δ ∈ F is a representative of the class Δ(q) ∈ F/℘(F). ([10, Prop. 5]) (ii) Let σ and τ be involutions in O(𝔼, φ) and O(V, q), respectively. Then there exist a, b ∈ F with a ≠ 0 such that φ ≃ [a, b] and (C(𝔼 ⊥ V, φ ⊥ q), J_{σ⊥τ}^{φ⊥q}) ≃ (C(𝔼′ ⊥ V, φ′ ⊥ a·q), J_{σ′⊥τ}^{φ′⊥a·q}) where (𝔼′, φ′) is the quadratic plane [a, b+a^{−1}δ], δ ∈ F is a representative of the class Δ(q) ∈ F/℘(F) and σ′ is a suitable involution in O(𝔼′, φ′). More precisely suppose that {x, y} is a standard symplectic basis of [a, b + a^{−1}δ] and t = dim Fix(𝔼 ⊥ V, σ ⊥ τ) − \frac{1}{2} dim(𝔼 ⊥ V). If t ≡ 0 mod 2 then σ′ = τ_{x} and if t ≡ 1 mod 2 then σ′ = id_{𝔼′}. In other words σ′(x) = x and σ′(y) = y + (t + 1)x^{−1}.

Proof. (i) This part has been proved in [10, Prop. 5], but as in the proof of part (ii) we need an explicit form of this isomorphism, we recall its construction here. Let {u, v} be a standard symplectic basis of (𝔼, φ = [a, b]), i.e., φ(u) = a, φ(v) = b and B_{φ}(u, v) = 1. Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a symplectic basis of (V, q) and set δ = ∑_{i=1}^{n} q(u_{i})q(v_{i}) ∈ F. Let {x, y} be a standard symplectic basis of (𝔼′, φ′ = [a, b + a^{−1}δ]). Define the linear map f : (𝔼 ⊥ V, φ ⊥ q) → C(𝔼′ ⊥ V, φ′ ⊥ a · q) via f(u) = x, f(v) = y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}), f(u_{i}) = x^{−1}u_{i} and f(v_{i}) = x^{−1}v_{i}, 1 ⩽ i ⩽ n. Let B = B_{φ⊥q} be the bilinear form associated to the quadratic form φ ⊥ q. We have the following relations:

f(u)^{2} = x^{2} = φ′(x) = a = φ(u),
f(u_{i})^{2} = (x^{−1}u_{i})^{2} = x^{−2}u_{i}^{2} = (a^{−1})(a · q(u_{i})) = q(u_{i}),
f(v_{i})^{2} = (x^{−1}v_{i})^{2} = x^{−2}v_{i}^{2} = (a^{−1})(a · q(v_{i})) = q(v_{i}).

Since {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} is a symplectic basis of (V, q), the set {u_{1}, a^{−1}v_{1}, ⋯, u_{n}, a^{−1}v_{n}} is also a symplectic basis of (V, a · q). We have a · q(u_{i}) = aq(u_{i}) and a·q(a^{−1}v_{i}) = a^{−1}q(v_{i}). So by (4.12) we have (the calculations are performed in C(V, a · q)):

(a^{−1}u_{1}v_{1} + ⋯ + a^{−1}u_{n}v_{n})^{2} = (a^{−1}u_{1}v_{1} + ⋯ + a^{−1}u_{n}v_{n})
+ aq(u_{1})a^{−1}q(v_{1}) + ⋯ + aq(u_{n})a^{−1}q(v_{n})
= a^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}) + δ,

So (u_{1}v_{1} +⋯+u_{n}v_{n})^{2} = a(u_{1}v_{1} +⋯+u_{n}v_{n})+a^{2}δ in C(V, a·q) and we obtain:

f(v)^{2} = (y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}))^{2}
= φ′(y) + a^{−2}x^{−2}(a(u_{1}v_{1} + ⋯ + u_{n}v_{n}) + a^{2}δ)
+ a^{−1}(x^{−1}y + yx^{−1})(u_{1}v_{1} + ⋯ + u_{n}v_{n})
= (b + a^{−1}δ) + a^{−3}(a(u_{1}v_{1} + ⋯ + u_{n}v_{n}) + a^{2}δ)
+ a^{−2}(xy + yx)(u_{1}v_{1} + ⋯ + u_{n}v_{n})
= b + a^{−1}δ + a^{−2}(u_{1}v_{1} + ⋯ + u_{n}v_{n}) + a^{−1}δ
+ a^{−2}(u_{1}v_{1} + ⋯ + u_{n}v_{n}) = b = φ(v).

Now clearly for every 1 ⩽ i ⩽ n, we have f(u)f(u_{i}) + f(u_{i})f(u) = 0 = B(u, u_{i}) and f(u)f(v_{i}) + f(v_{i})f(u) = 0 = B(u, v_{i}). Similarly for every 1 ⩽ i ≠ j ⩽ n, we have f(u_{i})f(u_{j}) + f(u_{j})f(u_{i}) = 0 = B(u_{i}, u_{j}), f(v_{i})f(v_{j}) + f(v_{j})f(v_{i}) = 0 = B(v_{i}, v_{j}) and f(u_{i})f(v_{j}) + f(v_{j})f(u_{i}) = 0 = B(u_{i}, v_{j}). For every 1 ⩽ i ⩽ n, we have

f(u_{i})f(v_{i}) + f(v_{i})f(u_{i}) = x^{−1}u_{i}x^{−1}v_{i} + x^{−1}v_{i}x^{−1}u_{i} = x^{−2}(u_{i}v_{i} + v_{i}u_{i})
= a^{−1}(a · B_{q}(u_{i}, v_{i})) = B(u_{i}, v_{i}).

We also have

f(u)f(v) + f(v)f(u) = x(y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}))
+ (y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}))x
= xy + yx = 1 = B(u, v).

For every 1 ⩽ i ⩽ n, we have

f(u_{i})f(v) + f(v)f(u_{i}) = x^{−1}u_{i}(y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}))
+ (y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}))x^{−1}u_{i}
= x^{−1}u_{i}y + yx^{−1}u_{i}
+ x^{−1}u_{i}(a^{−1}x^{−1}u_{i}v_{i}) + (a^{−1}x^{−1}u_{i}v_{i})x^{−1}u_{i}
= (a^{−1}x)u_{i}y + y(a^{−1}x)u_{i}
+ a^{−1}x^{−2}u_{i}(u_{i}v_{i}) + a^{−1}x^{−2}(u_{i}v_{i})u_{i}
= a^{−1}u_{i}(xy + yx) + a^{−2}(u_{i}u_{i}v_{i} + u_{i}(u_{i}v_{i} + a))
= a^{−1}u_{i} + a^{−1}u_{i} = 0 = B(u_{i}, v).

Similarly for every 1 ⩽ i ⩽ n, we have

f(v_{i})f(v) + f(v)f(v_{i}) = B(v_{i}, v).

So the map f is compatible with φ ⊥ q and it can be extended to an isomorphism f : C(𝔼 ⊥ V, φ ⊥ q) → C(𝔼′ ⊥ V, φ′ ⊥ a · q).

(ii) By (3.12), there is an anisotropic vector u ∈ 𝔼 such that σ(u) = u. (So if σ ≠ id_{𝔼}, then σ = τ_{u}). Extend {u} to a symplectic basis {u, v} of 𝔼. Set a = φ(u) and b = φ(v). Then φ ≃ [a, b]. By (i), we have C(𝔼 ⊥ V, φ ⊥ q) ≃ C(𝔼′ ⊥ V, φ′ ⊥ a · q) where (𝔼′, φ′) is the quadratic plane [a, b + a^{−1}δ] and δ ∈ F is a representative of the class Δ(q) ∈ F/℘(F). We claim that (C(𝔼 ⊥ V, φ ⊥ q), J_{σ⊥τ}^{φ⊥q}) ≃ (C(𝔼′ ⊥ V, φ′ ⊥ a · q), J_{σ′⊥τ}^{φ′⊥a·q}). We must show that for every w ∈ C(𝔼 ⊥ V) we have (f ∘ J_{σ⊥τ}^{φ⊥q})(w) = (J_{σ′⊥τ}^{φ′⊥a·q} ∘ f)(w). It is enough to check this for w = u, v, u_{i}, v_{i}, i = 1, ⋯, n. Note that in both cases σ′ = id_{𝔼′} and σ′ = τ_{x} we have σ′(x) = x. We have

(f ∘ J_{σ⊥τ}^{φ⊥q})(u) = f(u) = x = σ′(x) = J_{σ′⊥τ}^{φ′⊥a·q}(x)
= (J_{σ′⊥τ}^{φ′⊥a·q} ∘ f)(u).

Since for every 1 ⩽ i ⩽ n, τ(u_{i}) ∈ V, we get f(τ(u_{i})) = x^{−1}τ(u_{i}) and similarly f(τ(v_{i})) = x^{−1}τ(v_{i}). So for every 1 ⩽ i ⩽ n we have

(f ∘ J_{σ⊥τ}^{φ⊥q})(u_{i}) = f(τ(u_{i})) = x^{−1}τ(u_{i}) = σ′(x^{−1})τ(u_{i})
= J_{σ′⊥τ}^{φ′⊥a·q}(x^{−1}u_{i}) = (J_{σ′⊥τ}^{φ′⊥a·q} ∘ f)(u_{i}).

Similarly for every 1 ⩽ i ⩽ n we have

(f ∘ J_{σ⊥τ}^{φ⊥q})(v_{i}) = f(τ(v_{i})) = x^{−1}τ(v_{i}) = σ′(x^{−1})τ(v_{i})
= J_{σ′⊥τ}^{φ′⊥a·q}(x^{−1}v_{i}) = (J_{σ′⊥τ}^{φ′⊥a·q} ∘ f)(v_{i}).

Set t_{1} = dim Fix(𝔼, σ) − \frac{1}{2} dim 𝔼 and t_{2} = dim Fix(V, τ) − \frac{1}{2} dim V. Then t = t_{1}+t_{2}. Note that τ is also an involution in O(V, a·q, a·B_{q}). Since {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} is a τ-symplectic basis of (V, q), the set {u_{1}, a^{−1}v_{1}, ⋯, u_{n}, a^{−1}v_{n}} is also a τ-symplectic basis of (V, a · q). By (4.12), we have J_{τ}^{a·q}(u_{1}a^{−1}v_{1} + ⋯ + u_{n}a^{−1}v_{n}) = t_{2}·1+u_{1}(a^{−1}v_{1})+⋯+u_{n}(a^{−1}v_{n}) in C(V, a·q). So τ(v_{1})τ(u_{1})+⋯+ τ(v_{n})τ(u_{n}) = t_{2}·a+u_{1}v_{1}+⋯+u_{n}v_{n} in C(V, a·q). We have σ′(y) = y+(t+1)x^{−1}.

So using (3.13) we obtain

(J_{σ′⊥τ}^{φ′⊥a·q} ∘ f)(v) = J_{σ′⊥τ}^{φ′⊥a·q}(y + a^{−1}x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}))
= σ′(y) + a^{−1}(τ(v_{1})τ(u_{1}) + ⋯ + τ(v_{n})τ(u_{n}))σ′(x^{−1})
= y + (t + 1)x^{−1} + a^{−1}(t_{2} · a + u_{1}v_{1} + ⋯ + u_{n}v_{n})x^{−1}
= y + (t + t_{2} + 1)x^{−1} + a^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n})x^{−1}
= y + a^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n})x^{−1} + (t_{1} + 1)x^{−1}
= f(v + (t_{1} + 1)u^{−1}) = f(σ(v)) = (f ∘ J_{σ⊥τ}^{φ⊥q})(v),

which completes the proof.

Corollary 5.2. Let (V, q) be a 2n-dimensional quadratic space over a field F of characteristic 2 and let τ be a reflection of V. If n ⩾ 2, then there exists a 2ndimensional quadratic space (V′, q′) over F such that (C(V), J_{τ}) ≃ (C(V′), J_{id}).

Proof. Let u_{1} ∈ V be an anisotropic vector such that τ = τ_{u_{1}}. Extend {u_{1}} to a symplectic basis {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} of V. Set 𝔼_{1} = F u_{1} + F v_{1} and V_{1} = F u_{2} + F v_{2} + ⋯ + F u_{n} + F v_{n}. Then V = 𝔼_{1} ⊥ V_{1}, τ|_{𝔼_{1}} is a reflection and τ|_{V_{1}} = id. Set t := dim Fix(V, τ) − \frac{1}{2} dim V = (2n − 1) − n = n − 1. If n ≡ 0 mod 2, then by (5.1 (ii)), we have (C(V), J_{τ}) ≃ (C(𝔼_{1} ⊥ V_{1}), J_{τ|_{𝔼_{1}} ⊥id}) ≃ (C(V′), J_{id}) where (V′, q′) is a 2n-dimensional quadratic space. If n ≡ 1 mod 2, we have n ⩾ 3, so dim V_{1} ⩾ 4. As (V_{1}, q) is regular we can write V_{1} = 𝔼_{2} ⊥ V_{2} where 𝔼_{2} is a quadratic plane. We have dim Fix(𝔼_{1} ⊥ V_{2}, τ|_{𝔼_{1}} ⊥ id) − \frac{1}{2} dim(𝔼_{1} ⊥ V_{2}) = (2n − 3) − (n − 1) = n − 2 ≡ 1 mod 2. By (4.4) and (5.1 (ii)), we have

(C(V), J_{τ}) ≃ (C(𝔼_{1}), J_{τ|_{𝔼_{1}}} ) ⊗ (C(V_{1}), J_{id})
≃ ((C(𝔼_{1}), J_{τ|_{𝔼_{1}}} ) ⊗ (C(V_{2}), J_{id})) ⊗ (C(𝔼_{2}), J_{id})
≃ (C(V_{3}), J_{id}) ⊗ (C(𝔼_{2}), J_{id}) ≃ (C(V′), J_{id}).

Here V_{3} is a (2n − 2)-dimensional quadratic space which is determined by (5.1 (ii)) and V′ = V_{3} ⊥ 𝔼_{3}.

Corollary 5.3. Let F be a field of characteristic 2 and let (𝔼, φ) and (V, q) be quadratic spaces over F such that dim 𝔼 = 2. Let (𝔼′, φ′) be the quadratic plane [1, δ+δ′] where δ ∈ F is a representative of the class Δ(φ) ∈ F/℘(F) and δ′ ∈ F is a representative of the class Δ(q) ∈ F/℘(F). Let σ and τ be involutions in O(𝔼, φ) and O(V, q), respectively.

(i) We have C_{0}(𝔼 ⊥ V, φ ⊥ q) ≃ C_{0}(𝔼′, φ′) ⊗ C(V, a · q). ([10, Prop. 5]) (ii) We have (C_{0}(𝔼), J_{σ}) ⊗ (C(V), J_{τ}) ≃ (C_{0}(𝔼′ ⊥ V), J_{σ′⊥τ}) where σ′ is a suitable involution in O(𝔼′, φ′). (iii) There exists an element a ∈ D_{F} (φ) such that (C_{0}(𝔼 ⊥ V), J_{σ⊥τ}) ≃ (C_{0}(𝔼′), J_{σ′}) ⊗ (C(V, a · q), J_{τ}^{a·q}) where σ′ is a suitable involution in O(𝔼′, φ′).

More precisely if {x, y} is a standard symplectic basis of [1, δ + δ′] and t = dim Fix(𝔼 ⊥ V, σ ⊥ τ) − \frac{1}{2} dim(𝔼 ⊥ V), the involution σ′ ∈ O(𝔼′, φ′) in (ii) and (iii) can be described as follows: if t ≡ 0 mod 2 then σ′ = τ_{x} and if t ≡ 1 mod 2 then σ′ = id_{𝔼′}.

Proof. By (3.12), there is an anisotropic vector u ∈ 𝔼 such that σ(u) = u (so if σ ≠ id_{𝔼}, then σ = τ_{u}). Extend {u} to a symplectic basis {u, v} of 𝔼. Set a = φ(u) and b = φ(v). Then φ ≃ [a, b]. Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a τ-symplectic basis of V. Let δ = φ(u)φ(v) = ab ∈ F and δ′ = ∑_{i=1}^{n} q(u_{i})q(v_{i}) ∈ F.

(ii) By (4.5), (C_{0}(𝔼, φ), J_{σ}) ≃ (C_{0}(𝔼, a·φ), J_{σ}), so we may assume that a = 1, b = δ and φ = [1, δ]. By (5.1 (ii)), there exists an isomorphism (C(𝔼 ⊥ V, φ ⊥ q), J_{σ⊥τ}) ≃ (C(𝔼′ ⊥ V, φ′ ⊥ q), J_{σ′⊥τ}), such that if t ≡ 0 mod 2 then σ′ = τ_{x} and if t ≡ 1 mod 2 then σ′ = id_{𝔼′}. So σ′(x) = x and if σ′ ≠ id, then σ′ = τ_{x}. Using this isomorphism and (4.4), the map ψ : (C(𝔼, φ), J_{σ}) ⊗ (C(V, q), J_{τ}) ≃ (C(𝔼 ⊥ V, φ ⊥ q), J_{σ⊥τ}) ≃ (C(𝔼′ ⊥ V), J_{σ′⊥τ}) induced by ψ(u ⊗ 1) = x, ψ(v ⊗ 1) = y + x^{−1}(u_{1}v_{1} + ⋯ + u_{n}v_{n}), ψ(1 ⊗ u_{i}) = x^{−1}u_{i} and ψ(1 ⊗ v_{i}) = x^{−1}v_{i} is an isomorphism. The restriction of ψ to (C_{0}(𝔼), J_{σ}) ⊗ (C(V), J_{τ}) maps (C_{0}(𝔼), J_{σ}) ⊗ (C(V), J_{τ}) to (C_{0}(𝔼′ ⊥ V), J_{σ′⊥τ}) and the result is proved.

(iii) By (4.5), we have

(C_{0}(𝔼 ⊥ V), J_{σ⊥τ}^{φ⊥q}) ≃ (C_{0}(𝔼 ⊥ V), J_{σ⊥τ}^{a·φ⊥a·q}),  (1)

where (𝔼, a · φ) ≃ [1, δ]. Note that {a^{−1}u, v} is a standard symplectic basis of (𝔼, a · φ). Let (𝔼′′, φ′′) be the quadratic plane [1, δ + δ′ + δ′] ≃ [1, δ] ≃ (𝔼, a · φ) with a standard symplectic basis \{x′, y′\}. Set t_{1} = dim Fix(𝔼, σ)− \frac{1}{2} dim 𝔼, t_{2} = dim Fix(V, τ)− \frac{1}{2} dim V and t′_{1} = dim Fix(𝔼′, σ′)− \frac{1}{2} dim 𝔼′. We have t = t_{1} +t_{2}. By replacing 𝔼 with 𝔼′ and σ with σ′ in (ii), we obtain an isomorphism

(C_{0}(𝔼′), J_{σ′}) ⊗ (C(V), J_{τ}^{a·q}) ≃ (C_{0}(𝔼′′ ⊥ V), J_{σ′′⊥τ}^{φ′′⊥a·q}),  (2)

where σ′′ is an involution in O(𝔼′′, φ′′). Furthermore if t′_{1} + t_{2} ≡ 0 mod 2 then σ′′ = τ_{x′} and if t′_{1} + t_{2} ≡ 1 mod 2 then σ′′ = id_{𝔼′′}.

Now the linear map f : 𝔼 → 𝔼′′ defined by f(a^{−1}u) = x′ and f(v) = y′ induces an isomorphism C(𝔼, a · φ) ≃ C(𝔼′′, φ′′). Under this isomorphism, τ_{u} corresponds to τ_{x′} and id_{𝔼} corresponds to id_{𝔼′′}, i.e., σ corresponds to σ′′. So we can rewrite (2) as follows:

(C_{0}(𝔼′), J_{σ′}) ⊗ (C(V), J_{τ}^{a·q}) ≃ (C_{0}(𝔼 ⊥ V), J_{σ⊥τ}^{a·φ⊥a·q}),  (3)

Furthermore if t′_{1} + t_{2} ≡ 0 mod 2 then σ = τ_{u} (i.e., t_{1} = 0) and if t′_{1} + t_{2} ≡ 1 mod 2 then σ = id_{𝔼} (i.e., t_{1} = 1). So we have t′_{1} + t_{2} ≡ t_{1} mod 2 which implies that t = t_{1} + t_{2} ≡ t′_{1} mod 2.

Using (1) and (3) we obtain the desired isomorphism:

(C_{0}(𝔼 ⊥ V), J_{σ⊥τ}^{φ⊥q}) ≃ (C_{0}(𝔼′), J_{σ′}) ⊗ (C(V), J_{τ}^{a·q}).

Finally as t ≡ t′_{1} mod 2, if t ≡ 0 mod 2 then t′_{1} = 0 (i.e., σ′ = τ_{x}) and if t ≡ 1 mod 2 then t′_{1} = 1 (i.e., σ′ = id_{𝔼′}).

6 Multiquaternion algebras with involution

Let (Q, σ) be a quaternion algebra with involution of the first kind. In [9] it has been shown that if char F ≠ 2, then there exists a 2-dimensional quadratic space (V, q) and an involution τ in O(V, q) such that (Q, σ) ≃ (C(V), J_{τ}) (see [9, (6.2)]). Here we state this result for a field F of characteristic 2.

Theorem 6.1. Let (Q, σ) be a quaternion algebra with an involution of the first kind over a field F of characteristic 2. Then there exists a quadratic plane (𝔼, φ) over F with an involution τ ∈ O(𝔼, φ) such that (Q, σ) ≃ (C(𝔼), J_{τ}). Furthermore if σ is of symplectic type, then τ is necessarily equal to id and if σ is of orthogonal type, then τ ≠ id is necessarily a reflection.

Proof. Let Q = [α, β) for α ∈ F, β ∈ F^{×} and consider a standard basis {1, i, j, k} of Q. Let γ be the canonical involution of Q. Then by [14, Ch. 8, (7.4)], there exists some x ∈ Q^{∗} with γ(x) = x such that σ = Int(x) ∘ γ. So if we write x = a + bi + cj + dk for some a, b, c, d ∈ F, then b = 0 and we obtain x = a + cj + dk. If c = d = 0, then σ = γ. Let 𝔼 be the quadratic plane [β, αβ^{−1}] over F. By [14, Ch. 9, (4.6)], we have Q ≃ C(𝔼). Using (4.3), the involution J_{id} on C(𝔼) is of symplectic type. Since γ is the unique involution of symplectic type on Q, we have (Q, γ) ≃ (C(𝔼), J_{id}).

Now suppose that c or d is nonzero, then there exists s, t ∈ F such that cs + dt = a. Let (𝔼, φ) be a quadratic plane over F with a basis {u, v} and the relations φ(u) = t^{2} + αβ , φ(v) = s^{2} + β and B(u, v) = β ≠ 0. Define the linear map f : 𝔼 → Q by f(u) = t + k and f(v) = s + j. We have f(u)^{2} = (t + k)^{2} = t^{2} + αβ = φ(u) and f(v)^{2} = (s + j)^{2} = s^{2} + β = φ(v). We also have

f(u)f(v) + f(v)f(u) = (t + k)(s + j) + (s + j)(t + k) = kj + jk
= ijj + jij = (ij + ji)j = j^{2} = β = B(u, v).

So the map f is compatible with φ and it can be extended to an isomorphism ψ : C(𝔼) ≃ Q. Let I = ψ^{−1} ∘ γ ∘ ψ . Then I is an involution on C(𝔼) and (Q, γ) ≃ (C(𝔼), I). We have the following relations:

I(u) = ψ^{−1} ∘ γ ∘ ψ(u) = ψ^{−1} ∘ γ(t + k) = ψ^{−1}(t + k) = u,
I(v) = ψ^{−1} ∘ γ ∘ ψ(v) = ψ^{−1} ∘ γ(s + j) = ψ^{−1}(s + j) = v.

So I = J_{id_{𝔼}}. Now let T = ψ^{−1} ∘ σ ∘ ψ . Then T is an involution on C(𝔼) and (Q, σ) ≃ (C(𝔼), T). We have ψ(du + cv) = d(t + k) + c(s + j) = a + cj + dk = x, so ψ^{−1}(x) = du + cv ∈ 𝔼. Since σ = Int(x) ∘ γ, we obtain T = ψ^{−1} ∘ σ ∘ ψ = ψ^{−1} ∘ Int(x) ∘ γ ∘ ψ = (ψ^{−1} ∘ Int(x) ∘ ψ) ∘ (ψ^{−1} ∘ γ ∘ ψ) = Int(ψ^{−1}(x)) ∘ J_{id_{𝔼}}. By (4.1), in order to show that T = J_{τ} for some involution in O(𝔼, φ), it is enough to show that T (𝔼) = 𝔼 or equivalently T (w) = Int(ψ^{−1}(x)) ∘ J_{id_{𝔼}}(w) = Int(ψ^{−1}(x))(w) ∈ 𝔼 for every w ∈ 𝔼. But since x ∈ Q^{∗}, we get x^{2} ≠ 0, so φ(ψ^{−1}(x)) = ψ^{−1}(x)^{2} ≠ 0, i.e., ψ^{−1}(x) is anisotropic. On the other hand we have ψ^{−1}(x) ∈ 𝔼, so Int(ψ^{−1}(x))|_{𝔼} is a reflection and T (𝔼) = 𝔼. So σ = J_{τ} for some isometry τ of (𝔼, φ).

This proves the first part of the result. The second part follows from (4.3).

Corollary 6.2. Let (𝔼, φ, B) and (𝔼′, φ′, B′) be two quadratic planes over a field F of characteristic 2. If τ and τ′ are reflections of (𝔼, φ) and (𝔼′, φ′), respectively, then the following statements are equivalent: (1) (C(𝔼), J_{τ}) ≃ (C(𝔼′), J_{τ′}). (2) C(𝔼) ≃ C(𝔼′) and θ(τ) = θ(τ′).

Proof. According to (4.11), we have disc J_{τ} = θ(τ) and disc J_{τ′} = θ(τ′). The implication (1) ⇒ (2) is immediate and (2) ⇒ (1) follows from [6, (7.4)].

The next result is analogous to [9, (6.3)].

Theorem 6.3. Let F be a field of characteristic 2 and let (Q_{1}, σ_{1}), ⋯, (Q_{n}, σ_{n}) be quaternion algebras over F with involutions of the first kind. Let (A, σ) ≃ (Q_{1}, σ_{1}) ⊗ ⋯ ⊗ (Q_{n}, σ_{n}). Then there is a quadratic space (V, q) of dimension 2n over F and an involution τ in O(V, q) such that (A, σ) ≃ (C(V), J_{τ}). Furthermore if σ is of orthogonal type then dim Fix(V, τ) = \frac{1}{2} dim V and τ can be chosen to be an orthogonal sum of reflections and if σ is of symplectic type then τ can be chosen to be identity.

Proof. By (6.1), for each 1 ⩽ i ⩽ n there is a quadratic plane (𝔼_{i}, φ_{i}) with an involution τ_{i} ∈ O(𝔼_{i}, φ_{i}) such that (Q_{i}, σ_{i}) ≃ (C(𝔼_{i}), J_{τ_{i}}) and by (4.4), (A, σ) ≃ (C(V), J_{τ}) where (V, q) = (𝔼_{1}, φ_{1}) ⊥ ⋯ ⊥ (𝔼_{n}, φ_{n}) and τ = τ_{1} ⊥ ⋯ ⊥ τ_{n}. If σ is of orthogonal type then by [6, (2.23)], each σ_{i} is of orthogonal type and by (6.1), τ_{i} is a reflection for each i, so τ is an orthogonal sum of n reflections and dim Fix(V, τ) = \frac{1}{2} dim V. If σ is of symplectic type then by [6, (2.23)], at least one of the σ_{i}’s is of symplectic type, say σ_{1}. Then by (6.1), (Q_{1}, σ_{1}) ≃ (C(𝔼_{1}), J_{id}). Now if σ_{2} is of symplectic type then (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ≃ (C(𝔼_{1}), J_{id})⊗(C(𝔼_{2}), J_{id}) ≃ (C(𝔼_{1} ⊥ 𝔼_{2}), J_{id}). If σ_{2} is of orthogonal type, one can write (Q_{2}, σ_{2}) ≃ (C(𝔼_{2}), J_{τ_{2}}) where τ_{2} is a reflection of 𝔼_{2}. So (Q_{1}, σ_{1}) ⊗ (Q_{2}, σ_{2}) ≃ (C(𝔼_{1}), J_{id}) ⊗ (C(𝔼_{2}), J_{τ_{2}}) ≃ (C(𝔼_{1} ⊥ 𝔼_{2}), J_{id⊥τ_{2}}). By (5.2) this is isomorphic to (C(V), J_{id}) for a suitable 4-dimensional quadratic space (V, q) over F. Now induction on n completes the proof.

Lemma 6.4. Let K/F be a separable quadratic extension of fields of characteristic 2 with the nontrivial automorphism ρ. Then there is a quadratic plane (𝔼, φ) over F such that (K, ρ) ≃ (C_{0}(𝔼), J_{id}). Furthermore if (𝔼, φ) is a quadratic plane over F and τ is an involution in O(𝔼, φ) such that (K, ρ) ≃ (C_{0}(𝔼), J_{τ}), then τ is necessarily identity.

Proof. Since K/F is separable we may assume K = F (x) where x^{2}+x = δ ∈ F^{×}. Let (𝔼, φ) be the quadratic space [1, δ] with a standard symplectic basis {u, v}. Define f : K → C_{0}(𝔼) via f(λ + μx) = λ + μuv for every λ, μ ∈ F. We have f(x)^{2} = (uv)^{2} = uv + δ = f(x + δ) = f(x^{2}). So for every λ_{1}, μ_{1}, λ_{2}, μ_{2} ∈ F we have f((λ_{1} + μ_{1}x)(λ_{2} + μ_{2}x)) = f((λ_{1} + μ_{1}uv)(λ_{2} + μ_{2}uv)). Hence f is an F -algebra homomorphism and in fact an isomorphism, because it is obviously surjective. We have

J_{id_{𝔼}} ∘ f(x) = J_{id_{𝔼}}(uv) = vu = uv + 1 = f(x + 1) = f ∘ ρ(x).

So (K, ρ) ≃ (C_{0}(𝔼), J_{id}).

Now suppose that (K, ρ) ≃ (C_{0}(𝔼), J_{τ}) for a quadratic plane (𝔼, φ) and an involution τ ∈ O(𝔼, φ). As ρ(x) = x + 1, there is y ∈ C_{0}(𝔼) ⊆ C(𝔼) such that J_{τ} (y) = y +1, i.e., 1 ∈ Alt(C(𝔼), J_{τ}), so by [6, (2.6)], (C(𝔼), J_{τ}) is of symplectic type and hence by (4.3), τ = id.

Remark 6.5. In contrast with the case where char F ≠ 2, if (V, q) is a 2ndimensional quadratic space over a field F of characteristic 2, then there is no quadratic form q′ : V′ → F with dim V′ = 2n − 1 such that C_{0}(V) ≅ C(V′). In fact suppose that there is an F -algebra isomorphism f : C(V′) ≃ C_{0}(V). Since dim V′ is odd, (V′, q′) is not regular and there is an element u ∈ V′ such that B(u, v) = 0 for all v ∈ V′ and hence u ∈ Z(C(V′)). So we must have f(u) ∈ Z(C_{0}(V)). Therefore, we can write f(u) = λ + μ(u_{1}v_{1} + ⋯ + u_{n}v_{n}) where {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} is a symplectic basis of (V, q). Since f is an F -isomorphism, μ ≠ 0. On the other hand we have u^{2} = q′(u) ∈ F, but f(u)^{2} = λ^{2} + μ^{2}(q(u_{1})q(v_{1}) + ⋯ + q(u_{n})q(v_{n}) + u_{1}v_{1} + ⋯ + u_{n}v_{n}) ∉ F which is a contradiction.

Remark 6.6. Let K/F be a separable quadratic extension of fields with the nontrivial automorphism ρ. If char F ≠ 2, then there is a 1-dimensional quadratic space over F such that K ≃ C(V). But if char F = 2, then (6.4) and (6.5) show that there is no 1-dimensional quadratic space (V, q) over F such that K ≃ C(V).

Theorem 6.7. Let F be a field of characteristic 2 and let (Q_{1}, σ_{1}), ⋯, (Q_{n}, σ_{n}) be quaternion algebras over F with involutions of the first kind. Let K/F be a separable quadratic extension with the nontrivial automorphism ρ. If (A, σ) ≃ (Q_{1}, σ_{1})⊗⋯⊗(Q_{n}, σ_{n})⊗(K, ρ), then there is a quadratic space (V, q) of dimension 2n+2 over F and an involution τ in O(V, q) such that (A, σ) ≃ (C_{0}(V), J_{τ}). Furthermore if n ≡ 0 mod 2, then τ can be chosen to be identity and if n ≡ 1 mod 2, then τ can be chosen to be a reflection.

Proof. By (6.3), there is a quadratic space (V′, q′) of dimension 2n over F and an involution τ′ ∈ O(V′, q′) such that (Q_{1}, σ_{1})⊗⋯⊗(Q_{n}, σ_{n}) ≃ (C(V′), J_{τ′}). Also by (6.4), there is a quadratic plane (𝔼, φ) over F such that (K, ρ) ≃ (C_{0}(𝔼), J_{id}). By (6.3), we can choose (V′, τ′) such that either τ′ = id_{V′} or τ′ = τ′_{1} ⊥ ⋯ ⊥ τ′_{n} where τ′_{i}, i = 1, ⋯, n, is a reflection of some quadratic subplane 𝔼′_{i} of V′. Set t = dim Fix(V′ ⊥ 𝔼, τ′ ⊥ id) − \frac{1}{2} dim(V′ ⊥ 𝔼).

First suppose that τ′ = id_{V′}. We have t = n + 1. If n ≡ 0 mod 2, then by (5.3 (ii)), there exists a quadratic plane (𝔼′, φ′) over F such that (A, σ) ≃ (C(V′), J_{id}) ⊗ (C_{0}(𝔼), J_{id}) ≃ (C_{0}(V′ ⊥ 𝔼′), J_{id⊥id}). So (A, σ) ≃ (C_{0}(V), J_{id}) for V = V′ ⊥ 𝔼′. If n ≡ 1 mod 2, then by (5.3 (ii)), there exists a quadratic space (𝔼′, φ′) over F with a reflection τ′′ such that (A, σ) ≃ (C(V′), J_{id}) ⊗ (C_{0}(𝔼), J_{id}) ≃ (C_{0}(V′ ⊥ 𝔼′), J_{id⊥τ′′}). So (A, σ) ≃ (C_{0}(V), J_{τ}) for V = V′ ⊥ 𝔼′ and the reflection τ = id_{V′} ⊥ τ′′.

Now suppose that τ′ = τ′_{1} ⊥ ⋯ ⊥ τ′_{n}. We have t = 1, so by (5.3 (ii)), there exists a quadratic space (𝔼′, φ′) over F such that (A, σ) ≃ (C(V′), J_{τ′}) ⊗ (C_{0}(𝔼), J_{id}) ≃ (C_{0}(V′ ⊥ 𝔼′), J_{τ′⊥id_{𝔼′}}). If dim V′ = 2, then (A, σ) ≃ (C_{0}(V), J_{τ}) where V = V′ ⊥ 𝔼′ and τ = τ′ ⊥ id_{𝔼′} is a reflection of V. So suppose that dim V′ ⩾ 4. We have

(A, σ) ≃ (C_{0}(V′ ⊥ 𝔼′), J_{τ′⊥id_{𝔼′}}) ≃ (C_{0}(𝔼′_{1} ⊥ W), J_{τ′_{1}⊥ρ}),

where W = 𝔼′_{2} ⊥ ⋯ ⊥ 𝔼′_{n} ⊥ 𝔼′ and ρ = τ′_{2} ⊥ ⋯ ⊥ τ′_{n} ⊥ id_{𝔼′}. We have dim Fix(𝔼′_{1} ⊥ W, τ′_{1} ⊥ ρ) − \frac{1}{2} dim(𝔼′_{1} ⊥ W) = 1, so by (5.3 (iii)), (C_{0}(𝔼′_{1} ⊥ W), J_{τ′_{1}⊥ρ}) ≃ (C_{0}(𝔼′′), J_{id}) ⊗ (C(W, a · q_{1}), J_{ρ}) where (𝔼′′, φ′′) is a suitable quadratic space over F, q_{1} is the restriction of q′ ⊥ φ′ to W and a ∈ D_{F} (q|_{𝔼′_{1}}). Since dim Fix(W, ρ) = n + 1 > n = \frac{1}{2} dim W, by (4.7), the involution J_{ρ} on C(W) is of symplectic type. So by (6.3), there exists a quadratic space (V′′, q′′) over F such that (C(W, a · q_{1}), J_{ρ}) ≃ (C(V′′), J_{id}). So we have (A, σ) ≃ (C_{0}(𝔼′′), J_{id}) ⊗ (C(V′′), J_{id}). Now by an argument similar to the case τ′ = id_{V′}, we have (A, σ) ≃ (C_{0}(V), J_{id}) if n ≡ 0 mod 2, and (A, σ) ≃ (C_{0}(V), J_{τ}) for some reflection τ if n ≡ 1 mod 2 and the result is proved.

Corollary 6.8. Let (Q, σ) be a quaternion algebra with involution of the second kind over a field K of characteristic 2. Let F ⊆ K be the fixed subfield of σ|_{K}. Then there exists a 4-dimensional quadratic space (𝔸, q) over F such that (Q, σ) ≃ (C_{0}(𝔸), J_{τ}) where τ is a suitable reflection of (𝔸, q). ([5]).

Proof. The result follows from [14, Ch. 8, (11.2)] and (6.7).

Remark 6.9. With the notation of (6.7), if n ≡ 1 mod 2 then the involution τ ∈ O(V, q) can not be chosen to be identity; because in this case dim V = 2n + 2 ≡ 0 mod 4 and J_{id} is an involution on C(V) of the first kind, i.e., the involution σ is of the first kind which contradicts the assumption. In particular the involution τ of (6.8) can not be chosen to be identity. Also there is no (2n − 1)-dimensional quadratic space (V, q) over F such that A ≃ C(V). This fact is an easy consequence of (6.5) and (6.7).

Proposition 6.10. Let (𝔸, q) be a 4-dimensional quadratic space over a field F of characteristic 2 and let τ be an interchange isometry of (𝔸, q). Then (C(𝔸), J_{τ}) ≃ (C(ℍ_{1}), J_{τ_{u_{1}}} ) ⊗ (C(ℍ_{2}), J_{τ_{u_{2}}} ) where (ℍ_{1}, φ_{1}) and (ℍ_{2}, φ_{2}) are isometric to the hyperbolic plane [1, 0] over F with respective standard symplectic bases {u_{1}, v_{1}} and {u_{2}, v_{2}} and τ_{u_{i}}, i = 1, 2, is the reflection along the anisotropic vector u_{i} ∈ ℍ_{i}. In particular θ(τ_{u_{1}}) = θ(τ_{u_{2}}) = 1 ∈ F^{×}/F^{×2}. Furthermore if {w, x, y, z} is a basis of 𝔸 with the properties stated in (3.5), then (C(ℍ_{1}), J_{τ_{u_{1}}} ) (resp. (C(ℍ_{2}), J_{τ_{u_{2}}} )) is isomorphic, as an F -algebra with involution, to the F -quaternion algebra generated by {1 + x + w + xw, y + xz + xzw} (resp. {1 + x + xw, y + xz + xzw + z + zw}), which are invariant under J_{τ}.

Proof. Let {w, x, y, z} be a basis of 𝔸 with the properties stated in (3.5), i.e., q(w) = q(x) = q(y) = q(z) = 0, τ(w) = w, τ(x) = x, τ(y) = x + y, τ(z) = w + z, B(w, y) = B(x, z) = 1 and all other pairs of vectors from this basis are orthogonal. Set (𝔸′, q′, B′) = (ℍ_{1}, φ_{1}) ⊥ (ℍ_{2}, φ_{2}). Then the set B = {u_{1}, v_{1}, u_{2}, v_{2}} is a symplectic basis of 𝔸′. Define the linear map f : 𝔸′ → C(𝔸) by f(u_{1}) = 1 + x + w + xw, f(v_{1}) = y + xz + xzw, f(u_{2}) = 1 + x + xw and f(v_{2}) = y + xz + xzw + z + zw = f(v_{1}) + z + zw. Using the above relations and straightforward calculations in Clifford algebra we obtain that for every x, y ∈ B, we have f(x)^{2} = q′(x) and f(x)f(y) + f(y)f(x) = B′(x, y). So the map f is compatible with q′ and it can be extended to an isomorphism ψ : C(𝔸′) ≃ C(𝔸). We claim that for every v ∈ C(𝔸′), J_{τ} ∘ψ(v) = ψ∘(J_{τ_{u_{1}}} ⊥ J_{τ_{u_{2}}} )(v). It is enough to check this for v = u_{1}, v = v_{1}, v = u_{2} and v = v_{2}. We have the following relations:

J_{τ} ∘ ψ(u_{1}) = J_{τ} (1 + x + w + xw) = 1 + x + w + wx
= ψ(u_{1}) = ψ ∘ (J_{τ_{u_{1}}} ⊥ J_{τ_{u_{2}}} )(u_{1}),
J_{τ} ∘ ψ(u_{2}) = J_{τ} (1 + x + xw) = 1 + x + wx = ψ(u_{2}) = ψ ∘ (J_{τ_{u_{1}}} ⊥ J_{τ_{u_{2}}} )(u_{2}),
J_{τ} ∘ ψ(v_{1}) = J_{τ} (y + xz + xzw) = (x + y) + (w + z)x + w(w + z)x
= (x + y) + (wx + zx) + (q(w)x + wzx)
= (x + y) + (xw + xz + 1) + (0 + xzw + w)
= (y + xz + xzw) + (1 + x + w + xw) = ψ(v_{1} + u_{1})
= ψ(v_{1} + \frac{1}{q(u_{1})}u_{1}) = ψ ∘ (J_{τ_{u_{1}}} ⊥ J_{τ_{u_{2}}} )(v_{1}),

J_{τ} ∘ ψ(v_{2}) = J_{τ} (y + xz + xzw + z + zw)
= (x + y) + (w + z)x + w(w + z)x + (w + z) + w(w + z)
= (x + y) + (wx + zx) + (q(w)x + wzx) + (w + z) + (q(w) + wz)
= (x + y) + (xw + xz + 1) + (0 + xzw + w) + (w + z) + (0 + zw)
= (y + xz + xzw + z + zw) + (1 + x + xw) = ψ(v_{2} + u_{2})
= ψ(v_{2} + \frac{1}{q(u_{2})}u_{2}) = ψ ∘ (J_{τ_{u_{1}}} ⊥ J_{τ_{u_{2}}} )(v_{2}).

Finally note that θ(τ_{u_{1}}) = q(u_{1})F^{×2} = 1 ∈ F^{×}/F^{×2} and θ(τ_{u_{2}}) = q(u_{2})F^{×2} = 1 ∈ F^{×}/F^{×2} and the result is proved.

Remark 6.11. Using (6.10), one can give a shorter proof for (4.6) and (4.11 (b)).

7 Involutions of an even Clifford algebra with trivial Arf invariant

Remark 7.1. Let F be a field and let (A, σ) and (B, τ) be two F -algebras with involutions. Let A × B be the direct product of A and B. The involution on A × B defined by (a, b) ↦ (σ(a), τ(b)) is denoted by σ × τ.

The proof of the next result follows from the standard properties of tensor and direct product and we left it to the reader.

Lemma 7.2. Let F be a field and let (A, σ), (B, τ) and (C, ρ) be F -algebras with involution. Then there exists an isomorphism of F -algebras with involution ((A × B) ⊗_{F} C, (σ × τ) ⊗ ρ) ≃ ((A ⊗_{F} C) × (B ⊗_{F} C), (σ ⊗ ρ) × (τ ⊗ ρ)).

Remark 7.3. Let (V, q) be a quadratic space with trivial Arf invariant and let τ be an involution in O(V, q). We claim that there is a τ-symplectic basis {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} of V such that ∑_{i=1}^{n} q(u_{i})q(v_{i}) = 0. If τ is an orthogonal sum of interchange isometries, then by (3.6), V = ℍ_{1} ⊥ · · · · · · ℍ_{2s}. Choose a standard symplectic basis {u_{i}, v_{i}} of ℍ_{i} ≃ [1, 0], i = 1, ⋯, 2s. Then {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} is the desired basis. Otherwise there exists either an identity map or a reflection in the Wiitala decomposition of τ. So there is a τ-symplectic basis {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} of V such that q(u_{1}) ≠ 0. We have ∑_{i=1}^{n} q(u_{i})q(v_{i}) = a^{2} + a ∈ ℘(F), where a ∈ F. If a ≠ 0, by replacing v_{1} with v′_{1} = v_{1} + aq(u_{1})^{−1}u_{1}, we have ∑_{i=1}^{n} q(u_{i})q(v_{i}) = 0.

Lemma 7.4. For n ⩾ 2, let (𝔸_{1}, q_{1}), ⋯, (𝔸_{n}, q_{n}) be 4-dimensional hyperbolic spaces over a field F of characteristic 2. Then we have C_{0}(𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{n}) ≃ C_{0}(𝔸_{1}) ⊗ C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{n}).

Proof. For every 1 ⩽ i ⩽ n, there exist hyperbolic planes ℍ_{2i−1} and ℍ_{2i} such that 𝔸_{i} = ℍ_{2i−1} ⊥ ℍ_{2i}. Let {u_{i}, v_{i}}, i = 1, ⋯, 2n, be a standard symplectic basis of ℍ_{i} ≃ [1, 0]. Set (W_{1}, q_{1}) = ℍ_{1} ⊥ ℍ_{3} ⊥ ⋯ ⊥ ℍ_{2n−1} and (W_{2}, q_{2}) = ℍ_{2} ⊥ ℍ_{4} ⊥ ⋯ ⊥ ℍ_{2n}.

Define the linear map f_{1} : W_{1} → C(W_{1}) via f_{1}(u_{1}) = u_{1}, f_{1}(v_{1}) = v_{1} + u_{1}(u_{3}v_{3} + ⋯ + u_{2n−1}v_{2n−1}), f_{1}(u_{2i−1}) = u_{1}u_{2i−1} and f_{1}(v_{2i−1}) = u_{1}v_{2i−1} for i = 2, ⋯, n.

Similarly define the linear map f_{2} : W_{2} → C(W_{2}) via f_{2}(u_{2}) = u_{2}, f_{2}(v_{2}) = v_{2}+u_{2}(u_{4}v_{4}+⋯+u_{2n}v_{2n}), f_{2}(u_{2i}) = u_{2}u_{2i} and f_{2}(v_{2i}) = u_{2}v_{2i} for i = 2, ⋯, n.

By the proof of (5.1), the maps f_{1} and f_{2} are compatible with q_{1} and q_{2}, respectively; for example for the map f_{1}, considering the notation of the proof of (5.1) we have 𝔼′ = 𝔼 = ℍ_{1}, V = ℍ_{3} ⊥ ⋯ ⊥ ℍ_{2n−1}, u = x = u_{1}, v = y = v_{1}, a = q(u_{1}) = 1, x^{−1} = u_{1} and δ = q(u_{3})q(v_{3}) + ⋯ + q(u_{2n−1})q(v_{2n−1}) = 0. Now define the linear map f′_{1} : W_{1} → C(W_{1}) the same as f_{1} except that f′_{1}(v_{1}) = v_{1} + u_{1}(u_{3}v_{3} + ⋯ + u_{2n−1}v_{2n−1} + (n − 1) · 1) = f_{1}(v_{1}) + (n − 1) · u_{1}. We claim that f′_{1} is also compatible with q_{1}. It is enough to show that for every x, y ∈ {u_{1}, v_{1}, ⋯, u_{2n−1}, v_{2n−1}}, we have f′_{1}(x)^{2} = q_{1}(x) and f′_{1}(x)f′_{1}(y) + f′_{1}(y)f′_{1}(x) = B_{q_{1}}(x, y). It is clear that (n − 1) · u_{1} = f′_{1}(v_{1}) − f_{1}(v_{1}) commutes with f_{1}(u_{1}), f_{1}(u_{2i−1}) and f_{1}(v_{2i−1}) for i = 2, ⋯, n. Since f_{1} is compatible with q_{1}, in order to see that f′_{1} is compatible with q_{1}, it is enough to show that f′_{1}(v_{1})^{2} = q_{1}(v_{1}). We have

f′_{1}(v_{1})^{2} = (f_{1}(v_{1}) + (n − 1) · u_{1})^{2}
= f_{1}(v_{1})^{2} + (n − 1)^{2}q_{1}(u_{1}) · 1
+ (n − 1) · u_{1}(v_{1} + u_{1}(u_{3}v_{3} + ⋯ + u_{2n−1}v_{2n−1}))
+ (n − 1) · (v_{1} + u_{1}(u_{3}v_{3} + ⋯ + u_{2n−1}v_{2n−1}))u_{1}
= q_{1}(v_{1}) · 1 + (n − 1)^{2} · 1 + (n − 1) · (u_{1}v_{1} + v_{1}u_{1})
= q_{1}(v_{1}) · 1 + (n − 1) · 1 + (n − 1) · 1 = q_{1}(v_{1}) · 1.

So the claim is proved. Since the image of f′_{1} commutes with that of f_{2}, the map f = f′_{1} ⊕ f_{2} on W_{1} ⊥ W_{2} is compatible with q = q_{1} ⊥ q_{2} and it can be extended to an isomorphism f̄: C(L) = C(W_{1} ⊥ W_{2}) ≃ C(L) = C(𝔸_{1} ⊥ L′) where L = 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{n} and L′ = 𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{n}. By (4.4), there exists an isomorphism g : C(𝔸_{1} ⊥ L′) ≃ C(𝔸_{1}) ⊗ C(L′). So there exists an isomorphism ψ = g ∘f̄: C(L) ≃ C(𝔸_{1})⊗C(L′). The restriction of ψ to C_{0}(L) maps C_{0}(L) to C_{0}(𝔸_{1}) ⊗ C(L′). So we have an isomorphism ψ : C_{0}(L) ≃ C_{0}(𝔸_{1}) ⊗ C(L′).

Proposition 7.5. For n ⩾ 2, let (𝔸_{1}, q_{1}), ⋯, (𝔸_{n}, q_{n}) be 4-dimensional quadratic spaces over a field F of characteristic 2. For i = 1, ⋯, n, let τ_{i} be an interchange isometry of 𝔸_{i}. We have (C_{0}(𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{n}), J_{τ_{1}⊥···⊥τ_{n}}) ≃ (C_{0}(𝔸_{1}), J_{τ_{1}}) ⊗ (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{n}), J_{τ_{2}⊥···⊥τ_{n}}).

Proof. By (3.6), for every 1 ⩽ i ⩽ n, there exist quadratic planes ℍ_{2i−1} and ℍ_{2i} such that 𝔸_{i} = ℍ_{2i−1} ⊥ ℍ_{2i}, τ_{i}(ℍ_{2i−1}) = ℍ_{2i} and τ_{i}(ℍ_{2i}) = ℍ_{2i−1}. Let {u_{i}, v_{i}}, i = 1, ⋯, 2n, be a standard symplectic basis of ℍ_{i} ≃ [1, 0] such that for 1 ⩽ i ⩽ n, τ(u_{2i−1}) = u_{2i}, τ(u_{2i}) = u_{2i−1}, τ(v_{2i−1}) = v_{2i} and τ(v_{2i}) = v_{2i−1}. Set L = 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{n}, L′ = 𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{n}, τ = τ_{1} ⊥ ⋯ ⊥ τ_{n} and τ′ = τ_{2} ⊥ ⋯ ⊥ τ_{n}. Let ψ : C_{0}(L) ≃ C_{0}(𝔸_{1}) ⊗ C(L′) be the isomorphism discussed in (7.4). We want to show that (J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ = ψ ∘ J_{τ}. We have

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(u_{1}) = (J_{τ_{1}} ⊗ J_{τ′})(u_{1} ⊗ 1) = u_{2} ⊗ 1 = ψ(u_{2}) = ψ ∘ J_{τ} (u_{1}),

similarly

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(u_{2}) = ψ ∘ J_{τ} (u_{2}).

We also have

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(v_{1}) = (J_{τ_{1}} ⊗ J_{τ′})((n − 1) · u_{1} ⊗ 1 + v_{1} ⊗ 1
+ u_{1} ⊗ (u_{3}v_{3} + ⋯ + u_{2n−1}v_{2n−1}))
= (n − 1) · u_{2} ⊗ 1 + v_{2} ⊗ 1 + u_{2} ⊗ (v_{4}u_{4} + ⋯ + v_{2n}u_{2n})
= (n − 1) · u_{2} ⊗ 1 + v_{2} ⊗ 1
+ u_{2} ⊗ (u_{4}v_{4} + ⋯ + u_{2n}v_{2n} + (n − 1) · 1)
= v_{2} ⊗ 1 + u_{2} ⊗ (u_{4}v_{4} + ⋯ + u_{2n}v_{2n}) = ψ(v_{2}) = ψ ∘ J_{τ} (v_{1}),

similarly

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(v_{2}) = ψ ∘ J_{τ} (v_{2}).

For every 2 ⩽ i ⩽ n, we have

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(u_{2i−1}) = (J_{τ_{1}} ⊗ J_{τ′})(u_{1} ⊗ u_{2i−1}) = u_{2} ⊗ u_{2i}
= ψ(u_{2i}) = ψ ∘ J_{τ} (u_{2i−1}),

similarly

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(v_{2i−1}) = ψ ∘ J_{τ} (v_{2i−1}).

Finally for every 2 ⩽ i ⩽ n, we have

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(u_{2i}) = (J_{τ_{1}} ⊗ J_{τ′})(u_{2} ⊗ u_{2i}) = u_{1} ⊗ u_{2i−1}
= ψ(u_{2i−1}) = ψ ∘ J_{τ} (u_{2i}),

and similarly

(J_{τ_{1}} ⊗ J_{τ′}) ∘ ψ(v_{2i}) = ψ ∘ J_{τ} (v_{2i}).

Remark 7.6. Let (V, q) be a quadratic space over a field F of characteristic 2 and let C(V) be the Clifford algebra of (V, q). If τ is an involution in O(V, q), then τ induces an involution J_{τ}^{′q} on C(V) × C(V) such that J_{τ}^{′q}(v, 0) = (0, τ(v)) and J_{τ}^{′q}(0, v) = (τ(v), 0) for every v ∈ V. We usually denote J_{τ}^{′q} by J′_{τ}.

We need the following straightforward result.

Lemma 7.7. Let (V, q) and (V′, q′) be two quadratic spaces over a field F of characteristic 2 and let τ and τ′ be involutions in O(V, q) and O(V′, q′), respectively. If (C(V), J_{τ}) ≃ (C(V′), J_{τ′}), then (C(V) × C(V), J′_{τ}) ≃ (C(V′) × C(V′), J′_{τ′}).

Proposition 7.8. For n ⩾ 2, let (V, q) be a 2n-dimensional quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Suppose that Δ(q) is trivial and V = 𝔼 ⊥ V′ where 𝔼 is a quadratic subplane of V such that τ(𝔼) = 𝔼 and τ(V′) = V′. Let t = dim Fix(V, τ) − \frac{1}{2} dim V, τ′ = τ|_{V′} and u_{1} ∈ 𝔼 be an anisotropic vector such that τ(u_{1}) = u_{1}. Set a = q(u_{1}).

(a) If t ≡ 0 mod 2, then (C_{0}(V, q), J_{τ}) ≃ (C(V′, a·q)×C(V′, a·q), J_{τ′}^{a·q}×J_{τ′}^{a·q}).

(b) If t ≡ 1 mod 2, then (C_{0}(V, q), J_{τ}) ≃ (C(V′, a · q) × C(V′, a · q), J_{τ′}^{′a·q}).

Also C(V′, a · q) is Brauer equivalent to C(V, q). By abuse of notation, we have denoted the form a · q|_{V′} by a · q.

Proof. Let {u_{2}, v_{2}, ⋯, u_{n}, v_{n}} be a symplectic basis of V′. By (7.3), we can extend {u_{1}} to a symplectic basis {u_{1}, v_{1}} of 𝔼 such that q(u_{1})q(v_{1}) + ⋯ + q(u_{n})q(v_{n}) = 0. Let z = u_{1}v_{1} + ⋯ + u_{n}v_{n}. Then by [4, pp. 122-124], the map ψ : C(V′, a · q) × C(V′, a · q) → C_{0}(V, q) induced by ψ(u_{i}, 0) = u_{1}u_{i}z, ψ(v_{i}, 0) = u_{1}v_{i}z, ψ(0, u_{i}) = u_{1}u_{i}(z + 1) and ψ(0, v_{i}) = u_{1}v_{i}(z + 1), i = 1, ⋯, n, is an F -algebra isomorphism. Set σ = ψ^{−1} ∘ J_{τ} ∘ ψ. Then σ is an involution on C(V′, a · q) × C(V′, a · q).

(a) We must show that J_{τ} ∘ ψ = ψ ∘ (J_{τ′}^{a·q} × J_{τ′}^{a·q}) or equivalently σ = J_{τ′}^{a·q} × J_{τ′}^{a·q}. For this it suffices to verify that for every w, w′ ∈ C(V′) we have σ(w, w′) = (τ′(w), τ′(w′)). It is enough to check this for (w, w′) = (x, 0) and (w, w′) = (0, x) where x ∈ V′. By (4.12), we have τ(z) = z. As z ∈ Z(C_{0}(V)), we have zw = wz for every w ∈ C_{0}(V). Also u_{1}x = xu_{1} for every x ∈ V′. By the hypothesis, for every x ∈ V′ we have τ(x) ∈ V′ and

σ(x, 0) = ψ^{−1} ∘ J_{τ} ∘ ψ(x, 0) = ψ^{−1} ∘ J_{τ} (u_{1}xz) = ψ^{−1}(zτ(x)u_{1})
= ψ^{−1}(u_{1}τ(x)z) = (τ(x), 0) = (τ′(x), 0).

Similarly

σ(0, x) = ψ^{−1} ∘ J_{τ} ∘ ψ(0, x) = ψ^{−1} ∘ J_{τ} (u_{1}x(z + 1)) = ψ^{−1}((z + 1)τ(x)u_{1})
= ψ^{−1}(u_{1}τ(x)(z + 1)) = (0, τ(x)) = (0, τ′(x)).

So (C_{0}(V, q), J_{τ}) ≃ (C(V′, a · q) × C(V′, a · q), J_{τ′}^{a·q} × J_{τ′}^{a·q}).

(b) Like in the previous case, it suffices to verify that for every w, w′ ∈ C(V′) we have σ(w, w′) = (τ′(w′), τ′(w)). It is enough to check this for (w, w′) = (x, 0) and (w, w′) = (0, x) where x ∈ V′. By (4.12), we have τ(z) = z + 1. Again by the hypothesis, for every x ∈ V′ we have τ(x) ∈ V′ and

σ(x, 0) = ψ^{−1} ∘ J_{τ} ∘ ψ(x, 0) = ψ^{−1} ∘ J_{τ} (u_{1}xz) = ψ^{−1}((z + 1)τ(x)u_{1})
= ψ^{−1}(u_{1}τ(x)(z + 1)) = (0, τ(x)) = (0, τ′(x)).

Similarly

σ(0, x) = ψ^{−1} ∘ J_{τ} ∘ ψ(0, x) = ψ^{−1} ∘ J_{τ} (u_{1}x(z + 1)) = ψ^{−1}(zτ(x)u_{1})
= ψ^{−1}(u_{1}τ(x)z) = (τ(x), 0) = (τ′(x), 0).

So (C_{0}(V, q), J_{τ}) ≃ (C(V′, a · q) × C(V′, a · q), J_{τ′}^{′a·q}).

The last statement follows from (5.1 (i)) and the triviality of the Arf invariant.

Proposition 7.9. Let (V, q) be a 2n-dimensional quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Then there exists a quadratic space (V′, q′) such that (C(V) × C(V), J′_{τ}) ≃ (C(V′) × C(V′), J′_{id}).

Proof. If J_{τ} is of symplectic type, then by (6.3), there exists a quadratic space (V′, q′) over F such that (C(V), J_{τ}) ≃ (C(V′), J_{id}), so by (7.7), we have (C(V)× C(V), J′_{τ}) ≃ (C(V′) × C(V′), J′_{id}). So consider the case where J_{τ} is of orthogonal type, thus by (4.7), τ is either an orthogonal sum of reflections or an orthogonal sum of interchange isometries. By (6.10), the Clifford algebra of a 4-dimensional quadratic space with an involution induced by an interchange isometry is isomorphic to the Clifford algebra of a 4-dimensional quadratic space with an involution induced by an orthogonal sum of two reflections, so in order to prove the result we may consider the case where τ = τ_{1} ⊥ ⋯ ⊥ τ_{n} is an orthogonal sum of reflections. Let V = 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{n} be a Wiitala decomposition of (V, τ) such that τ_{i} = τ|_{𝔼_{i}}. Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a τ-symplectic basis of V and δ = q(u_{1})q(v_{1}) + ⋯ + q(u_{n})q(v_{n}) ∈ F. Let (𝔼_{n+1}, φ) be the quadratic plane [1, δ] over F with a standard symplectic basis {u_{n+1}, v_{n+1}} and set z = u_{1}v_{1}+⋯+u_{n+1}v_{n+1} ∈ C(V ⊥ 𝔼_{n+1}). Consider the involution ρ = τ ⊥ id_{𝔼_{n+1}} ∈ O(V ⊥ 𝔼_{n+1}, q ⊥ φ). We have dim Fix(V ⊥ 𝔼_{n+1}, ρ) − \frac{1}{2} dim(V ⊥ 𝔼_{n+1}) = (n + 2) − (n + 1) = 1 and q(u_{1})q(v_{1}) + ⋯ + q(u_{n+1})q(v_{n+1}) = 0, so by (7.8 (b)), we have (C_{0}(V ⊥ 𝔼_{n+1}), J_{ρ}) ≃ (C(V) × C(V), J′_{τ}); note that a = φ(u_{n+1}) = 1. On the other hand V ⊥ 𝔼_{n+1} = 𝔼_{1} ⊥ V_{1} where V_{1} = 𝔼_{2} ⊥ ⋯ ⊥ 𝔼_{n+1}. Again by (7.8 (b)), we have (C_{0}(V ⊥ 𝔼_{n+1}), J_{ρ}) ≃ (C_{0}(𝔼_{1} ⊥ V_{1}), J_{τ_{1}⊥ρ_{1}}) ≃ (C(V_{1}, a · q ⊥ a · φ) × C(V_{1}, a · q ⊥ a · φ), J′_{ρ_{1}}) where ρ_{1} = τ_{2} ⊥ ⋯ ⊥ τ_{n} ⊥ id_{𝔼_{n+1}} and a = q(u_{1}) ≠ 0. By (4.7), the involution J_{ρ_{1}} on C(V_{1}, a·q ⊥ a·φ) is of symplectic type, so by (6.3), we may write (C(V_{1}, a · q ⊥ a · φ), J_{ρ_{1}}) ≃ (C(V′, q′), J_{id}) for a suitable quadratic space (V′, q′). So by (7.7), (C(V_{1}, a · q ⊥ a · φ) × C(V_{1}, a · q ⊥ a · φ), J′_{ρ_{1}}) ≃ (C(V′) × C(V′), J′_{id}). Therefore, (C(V) × C(V), J′_{τ}) ≃ (C(V′) × C(V′), J′_{id}) and the result is proved.

Lemma 7.10. Let (𝔸, q) be a 4-dimensional quadratic space over a field F of characteristic 2 and let τ be an interchange isometry of (𝔸, q). Then (C_{0}(𝔸), J_{τ}) ≃ (C(ℍ), J_{τ_{u}}) × (C(ℍ), J_{id}) where ℍ is a hyperbolic subplane of 𝔸 and u is a suitable anisotropic element of ℍ.

Proof. By (3.6), there exist a decomposition 𝔸 = ℍ_{1} ⊥ ℍ_{2} of 𝔸 into hyperbolic planes ℍ_{1} and ℍ_{2} over F and standard symplectic bases {u_{i}, v_{i}} of ℍ_{i} ≃ [1, 0], i = 1, 2, such that τ(u_{1}) = u_{2}, τ(u_{2}) = u_{1}, τ(v_{1}) = v_{2} and τ(v_{2}) = v_{1}. We have q(u_{1})q(v_{1}) + q(u_{2})q(v_{2}) = 0. Set z = u_{1}v_{1} + u_{2}v_{2} ∈ C_{0}(𝔸). By (4.12), we have z^{2} = z and τ(z) = z. Also by (7.8), the map ψ : C(ℍ_{2}) × C(ℍ_{2}) → C_{0}(𝔸) induced by ψ(u_{2}, 0) = u_{1}u_{2}z, ψ(v_{2}, 0) = u_{1}v_{2}z, ψ(0, u_{2}) = u_{1}u_{2}(z + 1) and ψ(0, v_{2}) = u_{1}v_{2}(z + 1) is an isomorphism. Set σ = ψ^{−1} ∘ J_{τ} ∘ ψ. Then σ is an involution on C(ℍ_{2}) × C(ℍ_{2}). We claim that σ(x, y) = (J_{τ_{u_{2}}} (x), J_{id}(y)) for every x, y ∈ C(ℍ_{2}). It is enough to check this equality for (x, y) = (u_{2}, 0), (v_{2}, 0), (0, u_{2}) and (0, v_{2}). We have

σ(u_{2}, 0) = ψ^{−1} ∘ J_{τ} ∘ ψ(u_{2}, 0) = ψ^{−1} ∘ J_{τ} (u_{1}u_{2}z) = ψ^{−1}(zu_{1}u_{2})
= ψ^{−1}(u_{1}u_{2}z) = (u_{2}, 0) = (J_{τ_{u_{2}}} (u_{2}), 0),

Note that we have u_{1}z = u_{1}(u_{1}v_{1} + u_{2}v_{2}) = v_{1} + u_{1}u_{2}v_{2}, so v_{1} = u_{1}z + u_{1}u_{2}v_{2} and we obtain

σ(v_{2}, 0) = ψ^{−1} ∘ J_{τ} ∘ ψ(v_{2}, 0) = ψ^{−1} ∘ J_{τ} (u_{1}v_{2}z) = ψ^{−1}(zv_{1}u_{2})
= ψ^{−1}(u_{2}v_{1}z) = ψ^{−1}(u_{2}(u_{1}z + u_{1}u_{2}v_{2})z) = ψ^{−1}(u_{1}u_{2}z + u_{1}v_{2}z)
= (u_{2} + v_{2}, 0) = (v_{2} + \frac{B(u_{2}, v_{2})}{q(u_{2})}u_{2}, 0) = (J_{τ_{u_{2}}} (v_{2}), 0).

Similarly we have

σ(0, u_{2}) = ψ^{−1} ∘ J_{τ} ∘ ψ(0, u_{2}) = ψ^{−1} ∘ J_{τ} (u_{1}u_{2}(z + 1))
= ψ^{−1}((z + 1)u_{1}u_{2}) = ψ^{−1}(u_{1}u_{2}(z + 1)) = (0, u_{2}) = (0, J_{id}(u_{2})),

Finally

σ(0, v_{2}) = ψ^{−1} ∘ J_{τ} ∘ ψ(0, v_{2}) = ψ^{−1} ∘ J_{τ} (u_{1}v_{2}(z + 1)) = ψ^{−1}((z + 1)v_{1}u_{2})
= ψ^{−1}(u_{2}v_{1}(z + 1)) = ψ^{−1}(u_{2}(u_{1}z + u_{1}u_{2}v_{2})(z + 1))
= ψ^{−1}(0 + u_{1}v_{2}(z + 1)) = (0, v_{2}) = (0, J_{id}(v_{2})).

Theorem 7.11. Let (V, q) be a 2n-dimensional quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Suppose that Δ(q) is trivial and let t = dim Fix(V, τ) − \frac{1}{2} dim V.

(a) If t = 0 and τ is of reflectional kind, then there exists a (2n−2)-dimensional quadratic space (V′, q′) over F such that (C_{0}(V), J_{τ}) ≃ (C(V′)×C(V′), J_{τ′}× J_{τ′}) where τ′ is an involution in O(V′, q′) which is an orthogonal sum of n − 1 reflections. More precisely (V′, q′) is similar to a subform of (V, q) which can be explicitly described by (7.8).

(b) If t = 0 and τ is of interchanging kind, then there exist (2n−2)-dimensional quadratic spaces (V′, q′) and (V′′, q′′) over F such that (C_{0}(V), J_{τ}) ≃ (C(V′)×C(V′′), J_{τ′}×J_{id}) where τ′ is an involution in O(V′, q′) which is an orthogonal sum of n − 1 reflections. Furthermore we have C(V′) ≃ C(V′′) and C(V′) is isomorphic to the Clifford algebra of a hyperbolic space over F.

(c) If 0 ≠ t ≡ 0 mod 2, then there exists a (2n − 2)-dimensional quadratic space (V′, q′) over F such that (C_{0}(V), J_{τ}) ≃ (C(V′), J_{id}) × (C(V′), J_{id}).

(d) If t ≡ 1 mod 2, then there exists a (2n − 2)-dimensional quadratic space (V′, q′) over F such that (C_{0}(V), J_{τ}) ≃ (C(V′) × C(V′), J′_{id}).

Proof. Let V = W ⊥ 𝔼_{1} ⊥ ⋯ ⊥ 𝔼_{r} ⊥ 𝔸_{1} ⊥ ⋯ ⊥ 𝔸_{s} be a Wiitala decomposition of (V, τ) where W is a maximal fixed orthogonal summand of (V, τ). Let {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} be a τ-symplectic basis of V with respect to this decomposition. By (3.19), we have t = \frac{1}{2} dim W. Set z = u_{1}v_{1} + ⋯ + u_{n}v_{n} ∈ C_{0}(V). By (4.12), J_{τ} (z) = z + t · 1.

(a) We have W = 0 and J_{τ} (z) = z. As τ is of reflectional kind, we may assume that s = 0 and r = n, so one can write τ = τ_{u_{1}} ⊥ ⋯ ⊥ τ_{u_{n}} where τ_{u_{i}} is the reflection along the anisotropic vector u_{i} ∈ 𝔼_{i}. Let V′ = 𝔼_{2} ⊥ ⋯ ⊥ 𝔼_{n} and a = q(u_{1}). By (7.8 (a)), (C_{0}(V), J_{τ}) ≃ (C(V′, a · q) × C(V′, a · q), J_{τ′} × J_{τ′}) where τ′ = τ_{u_{2}} ⊥ ⋯ ⊥ τ_{u_{n}}.

(b) We have W = 0 and J_{τ} (z) = z. As τ is of interchanging kind, we may assume that r = 0, n = 2s and τ = τ_{1} ⊥ ⋯ ⊥ τ_{s} where τ_{i} is an interchange isometry of 𝔸_{i}, i = 1, ⋯, s. If n = 2, by (7.10), we have (C_{0}(V), J_{τ}) ≃ (C(ℍ)× C(ℍ), J_{τ′} × J_{id}) where ℍ is a hyperbolic subplane of V and τ′ is a reflection of ℍ. If n ⩾ 3, by (7.5), we have (C_{0}(V), J_{τ_{1}⊥···⊥τ_{s}}) ≃ (C_{0}(𝔸_{1}), J_{τ_{1}}) ⊗ (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{s}), J_{τ_{2}⊥···⊥τ_{s}}). Using (7.5), (7.10) and (7.2) we obtain

(C_{0}(V), J_{τ_{1}⊥···⊥τ_{s}}) ≃ (C_{0}(𝔸_{1}), J_{τ_{1}}) ⊗ (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{s}), J_{τ_{2}⊥···⊥τ_{s}})
≃ ((C(ℍ_{1}), J_{τ′_{1}}) × (C(ℍ_{1}), J_{id}))
⊗ (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{s}), J_{τ_{2}⊥···⊥τ_{s}})
≃ ((C(ℍ_{1}), J_{τ′_{1}}) ⊗ (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{s}), J_{τ_{2}⊥···⊥τ_{s}}))

× ((C(ℍ_{1}), J_{id}) ⊗ (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{s}), J_{τ_{2}⊥···⊥τ_{s}})),

where ℍ_{1} is a hyperbolic subplane of 𝔸_{1} and τ′_{1} is a reflection of ℍ_{1}. By (6.10), we have (C(𝔸_{2} ⊥ ⋯ ⊥ 𝔸_{s}), J_{τ_{2}⊥···⊥τ_{s}}) ≃ (C(L), J_{ρ}), where L = ℍ_{2} ⊥ ⋯ ⊥ ℍ_{2s−1} is an orthogonal sum of 2s − 2 = n − 2 hyperbolic planes ℍ_{i} over F and ρ = τ′_{2} ⊥ ⋯ ⊥ τ′_{2s−1} is an orthogonal sum of 2s − 2 reflections τ′_{i} of ℍ_{i}. So we obtain

(C_{0}(V), J_{τ_{1}⊥···⊥τ_{s}}) ≃ ((C(ℍ_{1}), J_{τ′_{1}}) ⊗ (C(L), J_{ρ})) × ((C(ℍ_{1}), J_{id}) ⊗ (C(L), J_{ρ}))
≃ (C(ℍ_{1} ⊥ L), J_{τ′_{1}⊥ρ}) × (C(ℍ_{1} ⊥ L), J_{id⊥ρ}).  (4)

By (4.7), (C(ℍ_{1} ⊥ L), J_{id⊥ρ}) is of symplectic type, so by (6.3), there exists a quadratic space (V′′, q′′) over F such that (C(𝔼′_{1} ⊥ L), J_{id⊥ρ}) ≃ (C(V′′), J_{id}), so (4) leads to (C_{0}(V), J_{τ}) ≃ (C(V′) × C(V′′), J_{τ′} × J_{id}), where V′ = ℍ_{1} ⊥ L and τ′ = τ′_{1} ⊥ ρ = τ′_{1} ⊥ ⋯ ⊥ τ′_{2s−1}. Finally note that as L is a hyperbolic space, V′ is also a hyperbolic space, so C(V′) ≃ C(V′′) is isomorphic to the Clifford algebra of a hyperbolic space over F.

(c) We have dim W ⩾ 4 and J_{τ} (z) = z. So we may assume that for the vector space W_{1} = F u_{1} + F v_{1} + F u_{2} + F v_{2} we have τ|_{W_{1}} = id_{W_{1}} where a = q(u_{1}) ≠ 0. Let 𝔼 = F u_{1} + F v_{1}, V_{1} = F u_{2} + F v_{2} + ⋯ + F u_{n} + F v_{n} and τ′ = τ|_{V_{1}}. By (7.8 (a)), (C_{0}(V), J_{τ}) ≃ (C(V_{1}, a · q) × C(V_{1}, a · q), J_{τ′} × J_{τ′}). On the other hand as dim Fix(V_{1}, τ|_{V_{1}}) − \frac{1}{2} dim V_{1} = t − 1 ⩾ 1, by (4.7), the involution J_{τ′} is of symplectic type, so by (6.3), we have (C(V_{1}), J_{τ′}) ≃ (C(V′), J_{id}) for some quadratic space (V′, q′) over F, so (C_{0}(V), J_{τ}) ≃ (C(V′) × C(V′), J_{id} × J_{id}).

(d) If t = 1, then dim W = 2 and we have V = W ⊥ W′ where W′ = W^{⊥}. Let u ∈ W be an anisotropic vector and a = q(u). By (7.8 (b)) we have (C_{0}(V, q), J_{τ}) ≃ (C_{0}(W ⊥ W′), J_{id_{W} ⊥τ′}) ≃ (C(W′, a · q) × C(W′, a · q), J′_{τ′}) where τ′ = τ|_{W′} and by (6.3) and (7.7), (C(W′, a · q) × C(W′, a · q), J′_{τ′}) ≃ (C(V′) × C(V′), J′_{id_{V ′}}) for some quadratic space (V′, q′). So (C_{0}(V, q), J_{τ}) ≃ (C(V′) × C(V′), J′_{id_{V ′}}).

Lemma 7.12. For i = 1, 2, let (A_{i}, σ_{i}) and (B_{i}, τ_{i}) be central simple algebras with involution over a field F. If (A_{1} × A_{2}, σ_{1} × σ_{2}) ≃ (B_{1} × B_{2}, τ_{1} × τ_{2}), then either (A_{1}, σ_{1}) ≃ (B_{1}, τ_{1}) and (A_{2}, σ_{2}) ≃ (B_{2}, τ_{2}) or (A_{1}, σ_{1}) ≃ (B_{2}, τ_{2}) and (A_{2}, σ_{2}) ≃ (B_{1}, τ_{1}).

Proof. Let ψ : (A_{1} × A_{2}, σ_{1} × σ_{2}) ≃ (B_{1} × B_{2}, τ_{1} × τ_{2}) be an isomorphism. The set B_{1} ×{0} is a two sided ideal of B_{1} ×B_{2}, so ψ^{−1}(B_{1} ×{0}) is a two sided ideal of A_{1} × A_{2}. As A_{1} and A_{2} are simple rings and ψ is an isomorphism, we have either ψ^{−1}(B_{1} × {0}) = A_{1} × {0} or ψ^{−1}(B_{1} × {0}) = {0} × A_{2}. Suppose that the first case occurs. Define a map f : A_{1} → B_{1} as follows: for every a_{1} ∈ A_{1}, there exists an element b_{1} ∈ B_{1} such that ψ(a_{1}, 0) = (b_{1}, 0). Set f(a_{1}) = b_{1}, i.e., for every a_{1} ∈ A_{1} we have ψ(a_{1}, 0) = (f(a_{1}), 0). As ψ : A_{1} × {0} → B_{1} × {0} is an isomorphism, the map f is an isomorphism of F -algebras. For every a_{1} ∈ A_{1} we have (τ_{1} × τ_{2}) ∘ ψ(a_{1}, 0) = ψ ∘ (σ_{1} × σ_{2})(a_{1}, 0), so (τ_{1} × τ_{2})(f(a_{1}), 0) = ψ(σ_{1}(a_{1}), 0). From this we obtain (τ_{1}(f(a_{1})), 0) = (f(σ_{1}(a_{1})), 0) which means that τ_{1} ∘ f = f ∘ σ_{1}. So we have proved that (A_{1}, σ_{1}) ≃ (B_{1}, τ_{1}). Similarly (A_{2}, σ_{2}) ≃ (B_{2}, τ_{2}).

If the second case occurs, a similar argument shows that (A_{1}, σ_{1}) ≃ (B_{2}, τ_{2}) and (A_{2}, σ_{2}) ≃ (B_{1}, τ_{1}) which completes the proof.

Corollary 7.13. For i = 1, ⋯, n, let (A_{i}, σ_{i}) and (B_{i}, τ_{i}) be central simple algebras with involution over a field F. If (A_{1} × ⋯ × A_{n}, σ_{1} × ⋯ × σ_{n}) ≃ (B_{1} × ⋯ × B_{n}, τ_{1} × ⋯ × τ_{n}), then there exists a permutation π ∈ S_{n} such that for every 1 ⩽ i ⩽ n, (A_{i}, σ_{i}) ≃ (B_{π(i)}, τ_{π(i)}).

Remark 7.14. The isomorphisms of (7.11 (a)) and (7.11 (b)) are the best possible, in the sense that the involution τ′ ∈ O(V′, q′) in (7.11 (a)) and (7.11 (b)) can not be chosen to be identity; because by (4.7), (C(V′), J_{τ′}) is of orthogonal type and (C(V′), J_{id}) is an algebra with involution of symplectic type. Now by (7.12), the involution J_{τ′} must be of orthogonal type, so τ′ can not be chosen to be identity.

In (7.11), we have shown that if τ ∈ O(V, q) is an involution and Δ(q) = 0, then (C_{0}(V), J_{τ}) can be decomposed as a direct product of two isomorphic Clifford algebras with involution. Conversely in (7.15) we will express a direct product of two isomorphic Clifford algebras with involution in the form of (C_{0}(V), J_{τ}) for a suitable quadratic space (V, q) with trivial Arf invariant and an involution τ ∈ O(V, q). The proof is very similar to that of (7.11).

Remark 7.15. For i = 1, 2, let (V_{i}, q_{i}) be a 2n-dimensional quadratic space over a field F of characteristic 2. For i = 1, 2, let τ_{i} be an involution in O(V_{i}, q_{i}) and set t_{i} = dim Fix(V_{i}, τ_{i}) − \frac{1}{2} dim V_{i}. Then there exists a (2n + 2)-dimensional quadratic space (V, q) over F with trivial Arf invariant such that (C(V_{1}) × C(V_{2}), J_{τ_{1}} × J_{τ_{2}}) ≃ (C_{0}(V), J_{τ}) if and only if one of the following conditions is satisfied:

(a) t_{1} = t_{2} = 0 and (C(V_{1}), J_{τ_{1}}) ≃ (C(V_{2}), J_{τ_{2}}). In this case τ is necessarily an orthogonal sum of reflections. In particular dim Fix(V, τ) = \frac{1}{2} dim V, so by (4.7), J_{τ} is necessarily of orthogonal type.

(b) t_{1} = 0 and 0 ≠ t_{2} ≡ 0 mod 2 (resp. t_{2} = 0 and 0 ≠ t_{1} ≡ 0 mod 2), n ≡ 1 mod 2, C(V_{1}) ≃ C(V_{2}) and for every u ∈ Fix(V_{1}, τ_{1}), we have q(u) ∈ F^{2}. In this case τ is necessarily an orthogonal sum of interchange isometries. In particular dim Fix(V, τ) = \frac{1}{2} dim V, so by (4.7), J_{τ} is necessarily of orthogonal type.

(c) 0 ≠ t_{i} ≡ 0 mod 2, i = 1, 2, and (C(V_{1}), J_{τ_{1}}) ≃ (C(V_{2}), J_{τ_{2}}). In this case τ has necessarily nontrivial maximal fixed orthogonal summand, i.e., dim Fix(V, τ) > \frac{1}{2} dim V, so by (4.7), J_{τ} is necessarily of symplectic type.

Lemma 7.16. Let (𝔼_{1}, φ_{1}) and (𝔼_{2}, φ_{2}) be two quadratic planes over a field F of characteristic 2. Let τ_{i} be a reflection of 𝔼_{i} and let τ′_{i} be an involution in O(𝔼_{i}, φ_{i}) where either (τ′_{1}, τ′_{2}) = (τ_{1}, τ_{2}) or (τ′_{1}, τ′_{2}) = (id, id). If (C(𝔼_{1}) × C(𝔼_{1}), J_{τ_{1}} × J_{τ′_{1}}) ≃ (C(𝔼_{2}) × C(𝔼_{2}), J_{τ_{2}} × J_{τ′_{2}}), then θ(τ_{1}) = θ(τ_{2}).

Proof. By the hypothesis we have (C(𝔼_{1}) × C(𝔼_{1}), J_{τ_{1}} × J_{τ_{1}}) ≃ (C(𝔼_{2}) × C(𝔼_{2}), J_{τ_{2}} × J_{τ_{2}}) or (C(𝔼_{1}) × C(𝔼_{1}), J_{τ_{1}} × J_{id}) ≃ (C(𝔼_{2}) × C(𝔼_{2}), J_{τ_{2}} × J_{id}). In the first case by (7.12), we have (C(𝔼_{1}), J_{τ_{1}}) ≃ (C(𝔼_{2}), J_{τ_{2}}) which implies that θ(τ_{1}) = θ(τ_{2}). So consider the second case. By (4.8 (b)), (C(𝔼_{1}), J_{τ_{1}}) is of orthogonal type and by (4.8 (a)), (C(𝔼_{2}), J_{id}) is of symplectic type, so (C(𝔼_{1}), J_{τ_{1}}) ≄ (C(𝔼_{2}), J_{id}). Again by (7.12), we have (C(𝔼_{1}), J_{τ_{1}}) ≃ (C(𝔼_{2}), J_{τ_{2}}), in particular θ(τ_{1}) = θ(τ_{2}).

Proposition 7.17. For n ⩾ 2, let (V, q) be a 2n-dimensional quadratic space over a field F of characteristic 2 and let τ be an involution in O(V, q). Suppose that Δ(q) is trivial and dim Fix(V, τ) = \frac{1}{2} dim V.

(i) If τ is of reflectional kind and (C_{0}(V), J_{τ}) ≃ (C(V′) × C(V′), J_{τ′} × J_{τ′}) is the isomorphism in (7.11 (a)), where (V′, q′) is a (2n−2)-dimensional quadratic space over F and τ′ is an involution in O(V′, q′) which is an orthogonal sum of reflections (in particular by (4.7), (C(V′), J_{τ′}) is of orthogonal type), then

(a) If n = 2 then disc J_{τ′} = θ(τ).

(b) If n ⩾ 3 then disc J_{τ′} = 1 ∈ F^{×}/F^{×2}.

(ii) If τ is of interchanging kind and (C_{0}(V), J_{τ}) ≃ (C(V′) × C(V′′), J_{τ′} × J_{id}) is the isomorphism of (7.11 (b)), where (V′, q′) and (V′′, q′′) are (2n − 2)- dimensional quadratic space over F and τ′ is an involution in O(V′, q′) which is an orthogonal sum of reflections (in particular by (4.7), (C(V′), J_{τ′}) is of orthogonal type), then disc J_{τ′} = 1 ∈ F^{×}/F^{×2}.

Proof. (i) (a) Note that by (4.11 (a)), disc J_{τ′} = θ(τ′) and by (7.16), θ(τ′) is independent of the choice of V′. As dim Fix(V, τ) = \frac{1}{2} dim V, (V, τ) has trivial maximal fixed orthogonal summand. Since τ is of reflectional kind, (V, τ) has a Wiitala decomposition V = 𝔼_{1} ⊥ 𝔼_{2} where τ_{i} = τ|_{𝔼_{i}} is a reflection along some anisotropic vector u_{i} ∈ 𝔼_{i}, i = 1, 2. By (7.8), we can write (C_{0}(V), J_{τ}) ≃ (C(𝔼_{2}, a · q) × C(𝔼_{2}, a · q), J_{τ_{2}} × J_{τ_{2}}) where a = q(u_{1}). By (4.11 (a)), disc J_{τ_{2}} = θ(τ_{2}) = aq(u_{2})F^{×2} = q(u_{1})q(u_{2})F^{×2} = θ(τ).

(b) We have dim V′ ⩾ 4, so by (4.11 (b)), disc J_{τ′} is trivial. (ii) If n = 2, by (7.10), we can choose (V′, q′) = (V′′, q′′) = (ℍ, φ) where (ℍ, φ) is a hyperbolic plane over F and τ′ = τ_{u} where φ(u) = 1. Also by (7.16), θ(τ′) is independent of the choice of 𝔼. So by (4.11), disc J_{τ′} = φ(u)F^{×2} = 1 ∈ F/F^{×2}. If n ⩾ 3, we have dim V′ ⩾ 4, so by (4.11 (b)), disc J_{τ′} is trivial.

A Appendix

Definition A.1. Let (V, q) be a quadratic space over a field F of characteristic 2 and let A be an F -algebra. A bilinear map ψ : V × V → A is called an even Clifford map if for every x, y, z ∈ V we have:

(a) ψ(x, y)ψ(y, z) = q(y) · ψ(x, z).

(b) ψ(x, x) = q(x) · 1_{A}.

Remark A.2. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let A be an F -algebra. Let ψ : V × V → A be an even Clifford map and u, v ∈ V. Then by computing ψ(u + v, u + v), we have ψ(u, v) + ψ(v, u) = B(u, v) · 1_{A}.

Lemma A.3. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let A be an F -algebra. Let ψ : V × V → A be an even Clifford map and u, v, w, x ∈ V. Then ψ(u, v)ψ(w, x) = ψ(u, w)ψ(v, x) + B(v, w) · ψ(u, x).

Proof. Using (a) in (A.1) and the bilinearity of ψ, we have

q(v+w)·ψ(u, x) = (q(v+w)+B(v, w))·ψ(u, x)+ψ(u, v)ψ(w, x)+ψ(u, w)ψ(v, x).

So ψ(u, v)ψ(w, x) = ψ(u, w)ψ(v, x) + B(v, w) · ψ(u, x).

Corollary A.4. Let (V, q, B) be a quadratic space over a field F of characteristic 2 and let A be an F -algebra. Let ψ : V × V → A be an even Clifford map and u, v ∈ V. Then ψ(u, v)ψ(u, v) = q(u)q(v) · 1_{A} + B(u, v) · ψ(u, v).

Theorem A.5. Let (V, q) be a quadratic space over a field F of characteristic 2. There exists an even Clifford map j from V × V to C_{0}(V) which satisfies the following conditions:

(a) As an F -algebra C_{0}(V) is generated by 1_{C_{0}(V)} and {j(x, y)|x, y ∈ V }.

(b) For every even Clifford map ψ : V × V → A, there exists a unique F - algebra homomorphism Ψ : C_{0}(V) → A such that ψ = Ψ ∘ j, i.e., the following diagram commutes:

\xymatrix{V × V \ar[dr]_{j} \ar[rr]^{ψ} &  & A \\  & C_{0}(V) \ar@{-->}[ur]_{Ψ}}  (5)

Proof. Define j : V × V → C_{0}(V) via j(u, v) = uv. Then j is clearly an even Clifford map. Let ψ : V × V → A be an even Clifford map. Let u_{1} ∈ V be an anisotropic vector and set a = q(u_{1}). Extend {u_{1}} to a symplectic basis {u_{1}, v_{1}, ⋯, u_{n}, v_{n}} of V and set z = u_{1}v_{1} + ⋯ u_{n}v_{n}. By (4.12), we have z^{2} = z + δ · 1 where δ = ∑_{i=1}^{n} q(u_{i})q(v_{i}) ∈ F is a representative of the class Δ(q) ∈ F/℘(F). By [4, (13.12)], the map g : C_{0}(V) → C(W, a · q) ⊗ (F + F z) induced by g(u_{1}w) = w ⊗ 1 and g(z) = 1 ⊗ z is an isomorphism, where W is the subspace of V generated by u_{2}, v_{2}, ⋯, u_{n}, v_{n}. Define the linear map f_{1} : W → A via f_{1}(w) = ψ(u_{1}, w). By (A.4), for every w ∈ W we have f_{1}(w)^{2} = ψ(u_{1}, w)ψ(u_{1}, w) = q(u_{1})q(w) · 1_{A} + B(u_{1}, w) · ψ(u_{1}, w) = a · q(w). So the map f_{1} is compatible with a · q and it can be extended to an F -algebra homomorphism f̄_{1} : C(W, a · q) → A. Now define the map f_{2} : F + F z → A via f_{2}(α + βz) = α + β(ψ(u_{1}, v_{1}) + ⋯ + ψ(u_{n}, v_{n})). By (A.4), we have ψ(u_{i}, v_{i})^{2} = q(u_{i})q(v_{i}) · 1_{A} + ψ(u_{i}, v_{i}). Also for every i ≠ j, by (A.2) and (A.3), we have

ψ(u_{i}, v_{i})ψ(u_{j}, v_{j}) = ψ(u_{i}, u_{j})ψ(v_{i}, v_{j}) = ψ(u_{j}, u_{i})ψ(v_{j}, v_{i}) = ψ(u_{j}, v_{j})ψ(u_{i}, v_{i})

So

f_{2}(z)^{2} = (ψ(u_{1}, v_{1}) + ⋯ + ψ(u_{n}, v_{n}))^{2} = ψ(u_{1}, v_{1})^{2} + ⋯ + ψ(u_{n}, v_{n})^{2}
= q(u_{1})q(v_{1}) · 1_{A} + ψ(u_{1}, v_{1}) + ⋯ + q(u_{n})q(v_{n}) · 1_{A} + ψ(u_{n}, v_{n})
= δ · 1_{A} + ψ(u_{1}, v_{1}) + ⋯ + ψ(u_{n}, v_{n}) = f_{2}(δ + z) = f_{2}(z^{2}).

Therefore, f_{2}((α_{1} + β_{1}z)(α_{2} + β_{2}z)) = f_{2}(α_{1} + β_{1}z)f_{2}(α_{2} + β_{2}z) for every α_{1}, β_{1}, α_{2}, β_{2} ∈ F and f_{2} is an F -algebra homomorphism.

We claim that the image of f̄_{1} commutes with that of f_{2}. We must show that for every w ∈ W:

ψ(u_{1}, w)(ψ(u_{1}, v_{1})+⋯+ψ(u_{n}, v_{n}))=(ψ(u_{1}, v_{1}) + ⋯ + ψ(u_{n}, v_{n}))ψ(u_{1}, w). (6)

It is enough to check this for w = u_{i} and w = v_{i}, i = 1, ⋯, n. If i = 1, (6) is clearly true. So fix an index i with 2 ⩽ i ⩽ n. If j ≠ 1 and j ≠ i, by (A.2) and (A.3), we have ψ(u_{1}, u_{i})ψ(u_{j}, v_{j}) = ψ(u_{j}, v_{j})ψ(u_{1}, u_{i}). If j = 1 or j = i, we have

ψ(u_{1}, u_{i})ψ(u_{1}, v_{1}) + ψ(u_{1}, u_{i})ψ(u_{i}, v_{i})
= ψ(u_{1}, u_{1})ψ(u_{i}, v_{1}) + B(u_{i}, u_{1}) · ψ(u_{1}, v_{1}) + q(u_{i}) · ψ(u_{1}, v_{i})
= q(u_{1}) · ψ(u_{i}, v_{1}) + q(u_{i}) · ψ(u_{1}, v_{i}).

Similarly

ψ(u_{1}, v_{1})ψ(u_{1}, u_{i}) + ψ(u_{i}, v_{i})ψ(u_{1}, u_{i}) = q(u_{1}) · ψ(u_{i}, v_{1}) + q(u_{i}) · ψ(u_{1}, v_{i}).

So ψ(u_{1}, u_{i})ψ(u_{1}, v_{1})+ψ(u_{1}, u_{i})ψ(u_{i}, v_{i}) = ψ(u_{1}, v_{1})ψ(u_{1}, u_{i})+ψ(u_{i}, v_{i}) ψ(u_{1}, u_{i}) and the equality (6) is achieved for w = u_{i}. Similarly (6) is true for w = v_{i} and the claim is proved. By [2, Ch. 5, Thm. 2] one can consider the F -algebra homomorphism f = f̄_{1} ⊗ f_{2} : C(W, a · q) ⊗ (F + F z) → A.

Set Ψ = f ∘ g. We claim that ψ = Ψ ∘ j. For every v ∈ V we can write v = αu_{1} + βv_{1} + w, where α, β ∈ F and w ∈ W. Since j(v, v′) = vv′ for every v, v′ ∈ V, it is enough to show that ψ(u_{1}, u_{1}) = Ψ(u_{1}u_{1}), ψ(v_{1}, v_{1}) = Ψ(v_{1}v_{1}), ψ(u_{1}, v_{1}) = Ψ(u_{1}v_{1}), ψ(u_{1}, w) = Ψ(u_{1}w), ψ(v_{1}, w) = Ψ(v_{1}w) and ψ(w_{1}, w_{2}) = Ψ(w_{1}w_{2}) for every w, w_{1}, w_{2} ∈ W. The first two relations follow from the F - linearity of Ψ and the property (b) of (A.1). For every w ∈ W we have

Ψ(u_{1}w) = f ∘ g(u_{1}w) = f̄_{1}(w) = ψ(u_{1}, w),

Similarly

Ψ(w_{1}w_{2}) = a^{−1} · Ψ(u_{1}w_{1}u_{1}w_{2}) = a^{−1} · (f ∘ g)(u_{1}w_{1})(f ∘ g)(u_{1}w_{2})
= a^{−1} · f̄_{1}(w_{1})f̄_{1}(w_{2}) = a^{−1} · ψ(u_{1}, w_{1})ψ(u_{1}, w_{2})
= a^{−1} · ψ(w_{1}, u_{1})ψ(u_{1}, w_{2}) = a^{−1}q(u_{1}) · ψ(w_{1}, w_{2}) = ψ(w_{1}, w_{2}).

The relations Ψ(u_{1}v_{1}) = ψ(u_{1}, v_{1}) and Ψ(wv_{1}) = ψ(w, v_{1}) are obtained with the same arguments. For the last one, we use wv_{1} = v_{1}w = a^{−1}(wu_{1}z + wu_{1}u_{2}v_{2} + ⋯ + wu_{1}u_{n}v_{n}).

Finally, the uniqueness of Ψ follows from the facts that as an F -algebra, C_{0}(V) is generated by 1_{C_{0}(V)} and the set {j(u, v)|u, v ∈ V }, and the relation Ψ(uv) = Ψ ∘ j(u, v) = ψ(u, v).

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M. G. Mahmoudi, mmahmoudi@sharif.ir, A.-H. Nokhodkar, anokhodkar@yahoo.com

Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11155-9415, Tehran, Iran.