On generic triality

M.-A. Knus, R. Parimala and R. Sridharan

November 15, 2000

Dedicated with gratitude to Professor K. Chandrasekharan on his 80^{th} birthday

1 Introduction

It is well known that the study of central simple algebras with involution is closely related to that of classical simple adjoint algebraic groups, the latter occuring as automorphism groups of these algebras (see for example [KMRT]). On the other hand there is a connection between central simple algebras with involution and algebraic groups via invariant theory. For each type of the simple groups A_{n}, B_{n} C_{n} and D_{n} there exists a generic algebra so that algebras in the corresponding class occur as specializations.The first case ever considered was by Amitsur and Procesi ([A], [P1]), who constructed a generic division algebra corresponding to the inner type of A_{n}. Generic central simple algebras with orthogonal or symplectic involutions were later introduced by Rowen [R1]. Procesi [P2] showed that for the groups PGL_{n}, PGO_{n} and PSp_{2n} the generic algebras occur as invariants of the groups acting on the matrix algebras over the field generated by the entries of the generic matrices. The centers of the generic algebras also occur as invariants of these groups acting on this field. As shown by Procesi [P2] for PGL_{n}, Berele and Saltman [BS] for PGO_{n} and PSp_{2n}, and more generally for any reductive group by Saltman [S1], the center can also be described as multiplicative invariant field of the corresponding Weyl group.

Let G be a simple adjoint connected algebraic group of type D_{4} over a field F of characteristic different from 2. A model is the group PGO^{+}(q) of proper similitudes of a nonsingular quadratic form q of dimension 8, modulo the center. The Dynkin diagram

[figure]

of G over a separable closure F̃ of F admits the permutation group of three symbols S_{3} as its automorphism group. It follows that S_{3} acts as outer automorphisms of G(F̃) and that AutF̃(G(F̃)) is the semidirect product G(F̃) ⋊ S_{3}. Following [KMRT] there is a notion of an associative trialitarian algebra T over F such that Aut(T) = G ⋊ S_{3} and G = Aut(T)^{0}, the connected component of identity of Aut(T). Very few examples of trialitarian algebras are known. In this paper, we construct generic trialitarian algebras on the lines of Amitsur, Rowen, Procesi and Saltman. We show that these algebras occur as invariants of the group PGO_{8}^{+} ⋊S_{3} acting on certain matrix algebras; the base field also occurs as invariants of the same group acting on the field generated by entries of generic matrices. We further show that the generic trialitarian algebra over a field F specializes to any given trialitarian algebra defined over an extension of F.

The paper is organized as follows. In Section 2 we recall some properties of the Clifford algebra of a nonsingular quadratic form q, which plays an important rôle in our construction, and show in some detail how it is related to the Lie algebra of the orthogonal group O(q). In Section 3 we describe triality at the level of Lie algebras (local triality), using the Clifford algebra of the norm form of a Cayley algebra, as in [KMRT]. In Section 4 we give an explicit description of the action of S_{3} on the Lie algebra of skew-symmetric (8×8)-matrices. Triality at the level of the adjoint algebraic group is the topic of Section 5. Some properties of central simple algebras with orthogonal involutions are recalled in Section 6. Trialitarian algebras are defined in Section 7. A cohomological definition of generic trialitarian algebras follows in Section 8 and it is shown that these algebras are division algebras. In the main Section 9 we give another, more explicit construction of generic trialitarian algebras, much in the spirit of the recent “Lectures on Division Algebras” [S2] of Saltman. The generic trialitarian algebra also occurs as the algebra of invariants of PGO_{8}^{+} ⋊S_{3} acting on generic matrices (Section 10). In Section 11 we show that the generic trialitarian algebra specializes to any trialitarian algebra. Similar constructions for generic algebras with orthogonal involutions with trivial discriminant, resp. with involutions of second kind, corresponding to the groups PGO_{2n}^{+}, resp. PGU_{n} or PGL_{n} ⋊ℤ/2ℤ are briefly sketched in the last Section 12.

Fields are assumed to have characteristic different from 2 and we restrict to commutative rings in which 2 is invertible. For any homomorphism of commutative rings ϕ : R → R′ and any R-module V, V ⊗_{ϕ} R′ denotes the tensor product of V with R′ over R. Unadorned tensor products are over F. For any ring A, A^{×} denotes the group of units of A.

Acknowledgement The authors thank both the Tata Institute of Fundamental Research, Mumbai, and the Eidgenössische Technische Hochschule, Zürich, for their respective hospitalities when this work was done.

2 The Clifford algebra and the Lie algebra of a quadratic space

Let q : V → F be a quadratic form on V, with associated polar bilinear form b_{q}(x, y) = q(x + y) − q(x) − q(y). We call the pair (V, q) a quadratic space if b_{q} is nonsingular. The adjoint involution σ_{q} on End_{F} (V) is defined through the identity

b_{q}(σ_{q}(f)(x), y) = b_{q}(x, f(y))

for x, y ∈ V and f ∈ End_{F} (V). The space

𝔬(q) = Skew( End_{F} (V), σ_{q})
= {f ∈ End_{F} (V) | σ_{q}(f) = −f}
= {f ∈ End_{F} (V) | b_{q}(x, f(y)) + b_{q}(f(x), y) = 0}

of skew-symmetric elements of End_{F} (V) with respect to the involution σ_{q} is a Lie subalgebra (of dimension \frac{n(n−1)}{2}) of End_{F} (V) for the Lie bracket [f, g] = f ∘ g − g ∘ f of End_{F} (V). The Lie algebra 𝔬(q) is the Lie algebra of the orthogonal group O(q). If q = < 1, 1, …, 1 >, i.e., q(∑_{i} x_{i}e_{i}) = ∑_{i} x_{i}^{2} with respect to a basis (e_{1}, …, e_{n}), and End_{F} (V) is identified with M_{n}(F) through the choice of the same basis, then σ_{q} is the transpose a ↦ a^{t} and 𝔬(q) = Skew_{n}(F), the set of (n × n)-skew-symmetric matrices with entries in F.

Let C(V, q) be the Clifford algebra of the quadratic space (V, q). We recall that C(V, q) = T V/I where T V is the tensor algebra of V and I is the ideal of T V generated by the elements x ⊗ x − q(x) · 1, x ∈ V. The vector space V, identified with a subspace of C(V, q) through the natural map V → C(V, q), generates C(V, q) as an algebra. The even Clifford algebra C_{0}(V, q) is the subalgebra of C(V, q) generated by products of an even number of elements of V. The properties of the Clifford algebra which we shall need are summarized in the following proposition (for a proof, see for example [SCH]):

Proposition 2.1 Let (V, q) be a nonsingular quadratic space of even dimension n = 2l.

1) The F -algebra C(V, q) is central simple of dimension 2^{2l} and has a unique involution τ which is the identity on V.

2) The center Z of C_{0}(V, q) is an étale quadratic algebra over F of the form Z = F [x]/(x^{2}−δ) where δ = (−1)^{l} det(b_{q}). The algebra C_{0}(V, q) is central separable over Z of rank 2^{2l−1}. Suppose that ℓ is even. Then the involution τ restricts to an involution τ_{0} of C_{0}(V, q) which is the identity on Z. Further τ_{0} is of orthogonal type if l is congruent to 0 modulo 4 and is of symplectic type if l is congruent to 2 modulo 4.

The Lie algebra 𝔬(q) can be identified with a subalgebra of Skew(C_{0}(V, q), τ_{0}), as follows. For x, y, z ∈ V and the Lie product [x, y] = xy − yx in C(V, q), we have:

[[x, y], z] = 2(b_{q}(y, z)x − b_{q}(x, z)y) ∈ V.  (1)

Let [V, V] ⊂ C(V, q) be the subspace spanned by the [x, y] = xy − yx for all x, y ∈ V. In view of (1) we have a linear map

ad: [V, V] → End_{F} (V), ξ ↦ ad_{ξ},

defined by ad_{ξ}(z) = [ξ, z] for ξ ∈ [V, V] and z ∈ V.

Lemma 2.2 The subspace [V, V] is a Lie subalgebra of Skew(C_{0}(V, q), τ_{0}), its image under ad is contained in 𝔬(q) and ad induces an isomorphism of Lie algebras:

ad: [V, V] ⟶^{∼} 𝔬(q).

Proof See [KMRT, Lemma (45.3)].

□

Example 2.3 Let q ⟶^{∼} < 1, 1, …, 1 > and let (e_{1}, …, e_{n}) be an orthonormal basis of q. Then [e_{i}, e_{j}] = 2e_{i}e_{j} and (e_{i}e_{j}, i < j), is a basis of [V, V]. Let (e_{ij}, 1 ≤ i, j ≤ n), be the standard basis of M_{n}(F), let E_{ij} be the skew-symmetric matrix e_{ij} − e_{ji}, i < j, and let f_{k} be the column vector with entry 1 in k-th position and zero entries elsewhere. We have

ad_{[e_{i},e_{j}]}(f_{k}) = 2 ad_{e_{i}e_{j}}(f_{k}) = 4(e_{i}δ_{jk} − e_{j}δ_{ik}) = 4E_{ij}f_{k}

so that ad^{−1} maps E_{ij} to the element \frac{1}{2}e_{i}e_{j} of C_{0}(V, q).

We have more in dimension 8:

Lemma 2.4 Let Z be the center of the even Clifford algebra C_{0}(V, q). If V has dimension 8, the embedding [V, V] ⊂ Skew(C_{0}(V, q), τ_{0}) induces a canonical isomorphism of Lie algebras over Z, [V, V] ⊗_{F} Z ⟶^{∼} Skew(C_{0}(V, q), τ_{0}). Thus ad induces a Z-isomorphism Skew(C_{0}(V, q), τ_{0}) ⟶^{∼} 𝔬(q) ⊗_{F} Z of Lie algebras.

Proof Since dim_{F} V = 8, the involution τ_{0} of C_{0}(V, q) is of orthogonal type (see (2.1)) and dim_{Z} Skew(C_{0}(V, q), τ_{0}) = 28. Let (e_{1}, …, e_{8}) be an orthogonal basis of V. The elements e_{i}e_{j}, i < j, are linearly independent over Z in C_{0}(V, q). Hence the canonical map [V, V] ⊗ Z → Skew(C_{0}(V, q), τ) induced by the product in C_{0}(V, q) is injective. It is surjective by a dimension count.

□

Similitudes of the quadratic space (V, q) are linear automorphisms of V with q(f(x)) = m(f)q(x), where m(f) ∈ F^{×} is the multiplier of the similitude. They form a group GO(q). If dim_{F} V = n is even, a similitude f is proper if det(f) = m(f)^{n/2}. Proper similitudes form a normal subgroup GO^{+}(q) of GO(q) of index 2. Similitudes are isometries if they have multiplier 1.

It readily follows from the definition of Clifford algebras that isometries of (V, q) induce automorphisms of C(V, q). For similitudes we have:

Proposition 2.5 Any similitude f ∈ GO(q) with multiplier m(f) induces an automorphism C(f) of (C_{0}(V, q), τ_{0}) such that C(f)(xy) = m(f)^{−1}f(x)f(y) for x, y ∈ V. The automorphism C(f) restricts to the identity on the center Z of C_{0}(V, q) if and only if f is proper. Further, we have ad ∘ C(f) ∘ ad^{−1} = Int(f).

Proof The first two claims are standard (see for example [KMRT]). Using the identity in (1) we have for x, y, z ∈ V:

(ad ∘ C(f))([x, y])(z) = 2m(f)^{−1}((f(x)b_{q}(f(y), z) − f(y)b_{q}(f(x), z))

and

\begin{aligned}(Int(f) ∘ ad)([x, y])(z) & = & 2f(xb_{q}(y, f^{−1}(z)) − yb_{q}(x, f^{−1}(z))) \\ & = & 2(f(x)b_{q}(y, f^{−1}(z)) − f(y)b_{q}(x, f^{−1}(z))). \end{aligned}

The last claim follows from m(f)^{−1}b_{q}(f(y), z) = b_{q}(y, f^{−1}(z)).

□

For any ν ∈ F^{×}, ν ·1_{V} is a similitude with multiplier ν^{2}, so that C(ν ·1_{V}) acts trivially on C_{0}(V, q) and PGO(q) = GO(q)/F^{×} acts on C_{0}(V, q). We observe that the homomorphism

C: PGO(q) → Aut_{F} (C_{0}(V, q), τ_{0})  (2)

is injective if dim V ≥ 3, in view of (2.5) and the fact that 𝔬(q) generates End_{F} (V) as an algebra. Let PGO^{+}(q) = GO^{+}(q)/F^{×}. By (2.5) the homomorphism (2) restricts to a homomorphism

C: PGO^{+}(q) → Aut_{Z}(C_{0}(V, q), τ_{0}).

3 Cayley algebras and local triality

The following presentation of local triality is taken from [KMRT]. Let ℭ be a Cayley algebra over F with conjugation π : x ↦ \overline{x} and norm 𝔫: x ↦ x\overline{x}. The new multiplication x ⋆ y = \overline{x} \overline{y} satisfies

x ⋆ (y ⋆ x) = (x ⋆ y) ⋆ x = 𝔫(x)y  (3)

for x, y ∈ ℭ. Further, the polar form b_{𝔫} is associative with respect to ⋆, in the sense that

b_{𝔫}(x ⋆ y, z) = b_{𝔫}(x, y ⋆ z).

Proposition 3.1 For x, y ∈ ℭ, let r_{x}(y) = y⋆x and ℓ_{x}(y) = x⋆y. The map ℭ → End_{F} (ℭ⊕ℭ) given by

x ↦ (\begin{matrix}0 & ℓ_{x} \\ r_{x} & 0 \end{matrix})

induces isomorphisms α: (C(ℭ, 𝔫), τ) ⟶^{∼} (End_{F} (ℭ ⊕ ℭ), σ_{𝔫⊥𝔫}) and

α_{0} : (C_{0}(ℭ, 𝔫), τ_{0}) ⟶^{∼} (End_{F} (ℭ), σ_{𝔫}) × (End_{F} (ℭ), σ_{𝔫}),  (4)

of algebras with involution. Further α_{0} maps [ℭ, ℭ] ⊂ C_{0}(ℭ, 𝔫) into 𝔬(𝔫) × 𝔬(𝔫).

Proof We have r_{x}(ℓ_{x}(y)) = ℓ_{x}(r_{x}(y)) = 𝔫(x) · y by (3). Thus the existence of the map α follows from the universal property of the Clifford algebra. The fact that α is compatible with involutions is equivalent to

b_{𝔫}(x ⋆ (z ⋆ y), u) = b_{𝔫}(z, y ⋆ (u ⋆ x))

for all x, y, z, u in ℭ. This formula follows from

b_{𝔫}(x ⋆ (z ⋆ y), u) = b_{𝔫}(u, x ⋆ (z ⋆ y)) = b_{𝔫}(u ⋆ x, z ⋆ y) = b_{𝔫}(z ⋆ y, u ⋆ x) = b_{𝔫}(z, y ⋆ (u ⋆ x)).

Since C(ℭ, 𝔫) is central simple, the map α is an isomorphism by a dimension count. The fact that the image of [ℭ, ℭ] under α_{0} lies in 𝔬(𝔫) × 𝔬(𝔫) follows from the fact that α_{0} is an isomorphism of algebras with involution.

□

We have an (injective) homomorphism

α_{0}|_{[ℭ,ℭ]} ∘ ad^{−1} : 𝔬(𝔫) → 𝔬(𝔫) × 𝔬(𝔫) ⊂ End_{F} (ℭ) × End_{F} (ℭ).

For any λ ∈ 𝔬(𝔫) let α_{0}|_{[ℭ,ℭ]} ∘ ad^{−1}(λ) = (λ^{+}, λ^{−}).

Proposition 3.2 (Local triality) For any λ ∈ 𝔬(𝔫), the elements λ^{+}, λ^{−} ∈ 𝔬(𝔫) satisfy

λ^{+}(x ⋆ y) = λ(x) ⋆ y + x ⋆ λ^{−}(y),  (5)
λ^{−}(x ⋆ y) = λ^{+}(x) ⋆ y + x ⋆ λ(y),  (6)
λ(x ⋆ y) = λ^{−}(x) ⋆ y + x ⋆ λ^{+}(y)  (7)

for all x, y ∈ ℭ. Further the pair (λ^{+}, λ^{−}) is uniquely determined by any of the three relations.

Proof Let ξ = ad^{−1}(λ). Since α_{0} is an isomorphism of algebras we have

α_{0}([ξ, x]) = [α_{0}(ξ), α_{0}(x)] = (\begin{matrix}λ^{+} & 0 \\ 0 & λ^{−} \end{matrix}) (\begin{matrix}0 & ℓ_{x} \\ r_{x} & 0 \end{matrix}) − (\begin{matrix}0 & ℓ_{x} \\ r_{x} & 0 \end{matrix}) (\begin{matrix}λ^{+} & 0 \\ 0 & λ^{−} \end{matrix}) .

On the other hand, [ξ, x] = λ(x) and

α_{0}([ξ, x]) = (\begin{matrix}0 & ℓ_{λx} \\ r_{λx} & 0 \end{matrix})

so that

λ^{+}(x ⋆ y) − x ⋆ λ^{−}(y) = λ(x) ⋆ y
λ^{−}(y ⋆ x) − λ^{+}(y) ⋆ x = y ⋆ λ(x).

This gives formulas (5) and (6). From (5) we obtain

b_{𝔫}(λ^{+}(x ⋆ y), z) = b_{𝔫}(λ(x) ⋆ y, z) + b_{𝔫}(x ⋆ λ^{−}(y), z).

Since b_{𝔫}(x ⋆ y, z) = b_{𝔫}(x, y ⋆ z) and since λ^{−}, λ and λ^{+} are in 𝔬(𝔫), this implies

−b_{𝔫}(x, y ⋆ λ^{+}(z)) = −b_{𝔫}(x, λ(y ⋆ z)) + b_{𝔫}(x, λ^{−}(y) ⋆ z)

for all x, y, and z in 𝔬(𝔫), hence (7). We finally check that the pair (λ^{+}, λ^{−}) associated to λ is uniquely determined by (5). It suffices to check that the only pair of linear maps λ_{1}, λ_{2} ∈ 𝔬(𝔫) satisfying λ_{1}(x ⋆ y) = x ⋆ λ_{2}(y) for all x, y ∈ ℭ is the pair (0, 0). We have λ_{1}(\overline{x} \overline{y}) = \overline{x}\overline{λ_{2}(y)}. Then x = 1 implies λ_{1}(\overline{y}) = \overline{λ_{2}(y)}, so that λ_{1}(\overline{x} \overline{y}) = \overline{x}λ_{1}(\overline{y}) and λ_{1}(x) = xa for a = λ_{1}(1). This finally implies (xy)a = x(ya) for all x, y ∈ ℭ and a lies in F. However λ_{1}(x) = ax, for a ∈ F and λ_{1} ∈ 𝔬(𝔫), implies a = 0.

□

Let d_{ρ}, resp. d_{ρ^{2}} be the endomorphisms of 𝔬(𝔫) defined by d_{ρ}(λ) = λ^{+} and d_{ρ^{2}}(λ) = λ^{−} for λ ∈ 𝔬(𝔫), so that

α_{0} ∘ ad^{−1} = (d_{ρ}, d_{ρ^{2}}) ∈ End_{F} (ℭ) × End_{F} (ℭ).

Corollary 3.3 The endomorphisms d_{ρ} and d_{ρ^{2}} are Lie algebra automorphisms of 𝔬(𝔫) and satisfy the relations (d_{ρ})^{2} = d_{ρ^{2}}, (d_{ρ})^{3} = 1.

Proof The claims follow from uniqueness in (3.2).

□

The conjugation π of ℭ induces an automorphism (of Lie algebras) d_{π} : λ ↦ πλπ of 𝔬(𝔫) (the product πλπ is taken in End_{F} (ℭ)).

Proposition 3.4 The relations (d_{π})^{2} = 1 and d_{π} ∘ d_{ρ} = d_{ρ^{2}} ∘ d_{π} hold in Aut_{F} (𝔬(𝔫)) and {d_{π}, d_{ρ}} generate a subgroup of Aut_{F} (𝔬(𝔫)), isomorphic to S_{3}.

Proof The first relation is obvious. We check the second one. Since π is an isometry of 𝔫, π induces an automorphism C(π) of C(ℭ, 𝔫). We have C(π) = Int(e) for e = 1_{ℭ} ∈ ℭ, hence α ∘ C(π) ∘ α^{−1} = Int(α(e)). Plugging in the definition of α, we get for ( \begin{matrix}f & 0 \\ 0 & g \end{matrix} ) ∈ End_{F} (ℭ ⊕ ℭ),

(α ∘ C(π) ∘ α^{−1})( \begin{matrix}f & 0 \\ 0 & g \end{matrix} ) = ( \begin{matrix}πgπ & 0 \\ 0 & πfπ \end{matrix} ).  (8)

On the other hand we know that ad ∘ C(π) ∘ ad^{−1} = Int(π) on 𝔬(𝔫) by (2.5), so that

(α_{0} ∘ C(π) ∘ α_{0}^{−1})(d_{ρ}(λ), d_{ρ^{2}}(λ)) = (πd_{ρ^{2}}(λ)π, πd_{ρ}(λ)π)
= (α_{0} ∘ C(π) ∘ ad^{−1})(λ)
= (α_{0} ∘ ad^{−1} ∘ ad ∘ C(π) ∘ ad^{−1})(λ)
= (α_{0} ∘ ad^{−1})(πλπ)
= (d_{ρ}(πλπ), d_{ρ^{2}}(πλπ))

and the second relation holds. Thus we get a homomorphism S_{3} → Aut_{F} (𝔬(𝔫)). The fact that it is injective follows from the explicit formulas given in the next section.

□

Remark 3.5 The elements λ of 𝔬_{8} fixed under the action of S_{3} are such that λ(x ⋆ y) = λ(x)⋆y +x⋆λ(y) for all x, y ∈ ℭ. Such λ are derivations of (ℭ, ⋆). They define a Lie algebra of type G_{2}.

4 Triality for Skew-symmetric Matrices

Let q be the 8-dimensional quadratic form < 1, 1, …, 1 >. The Lie algebra of skewsymmetric (8 × 8)-matrices is the Lie algebra 𝔬_{8} = 𝔬(q). In this section we give explicit formulas for the trialitarian action on 𝔬_{8}. We call the Cayley algebra with norm the identity form < 1, 1, …, 1 > the standard Cayley algebra or the algebra of octonions and denote it ℭ(8). There exists an orthogonal basis (1_{ℭ} = e_{1}, …, e_{8}) of ℭ(8) such that the multiplication table of ℭ(8) is

\begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} & e_{8} \\ e_{1} & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} & e_{8} \\ e_{2} & e_{2} & −e_{1} & e_{4} & −e_{3} & −e_{6} & e_{5} & −e_{8} & e_{7} \\ e_{3} & e_{3} & −e_{4} & −e_{1} & e_{2} & −e_{7} & e_{8} & e_{5} & −e_{6} \\ e_{4} & e_{4} & e_{3} & −e_{2} & −e_{1} & −e_{8} & −e_{7} & e_{6} & e_{5} \\ e_{5} & e_{5} & e_{6} & e_{7} & e_{8} & −e_{1} & −e_{2} & −e_{3} & −e_{4} \\ e_{6} & e_{6} & −e_{5} & −e_{8} & e_{7} & e_{2} & −e_{1} & −e_{4} & e_{3} \\ e_{7} & e_{7} & e_{8} & −e_{5} & −e_{6} & e_{3} & e_{4} & −e_{1} & −e_{2} \\ e_{8} & e_{8} & −e_{7} & e_{6} & −e_{5} & e_{4} & −e_{3} & e_{2} & −e_{1} \end{aligned}  (9)

Let x_{ij}, 1 ≤ i, j ≤ 8, be indeterminates and let F (x_{ij}) be the quotient field of the polynomial ring F [x_{ij}] in the indeterminates x_{ij}. The matrix X = ∑_{i,j} x_{ij}e_{ij} ∈ M_{8}(F (x_{ij})) is the generic matrix and the matrix X = ∑_{i<j} x_{ij}E_{ij} ∈ 𝔬_{8} ⊗ F (x_{ij}), E_{ij} = e_{ij} − e_{ji}, is the generic skew-symmetric matrix. We compute the image of X under the automorphisms d_{ρ} and d_{π}. The element E_{ij} corresponds to the product \frac{1}{2}e_{i}e_{j} in the Clifford algebra C(8), through the map ad: [ℭ(8), ℭ(8)] → 𝔬(8) (see (2.3)). Thus the image of E_{ij} under d_{ρ} is the matrix of the automorphism u ↦ \frac{1}{2}e_{i} ⋆ (u ⋆ e_{j}) = \frac{1}{2} \overline{e_{i}} · (e_{j} · u) of the space ℭ. Straightforward explicit calculations using the multiplication table (9) show that X = ∑_{i<j} x_{ij}E_{ij} has as image under α_{0} ∘ ad^{−1} the pair of skew-symmetric matrices (d_{ρ}(X), d_{ρ^{2}}(X)), where

d_{ρ}(X) = \frac{1}{2} ( \begin{matrix}−x_{12} + x_{34} & −x_{13} − x_{24} & −x_{14} + x_{23} & −x_{15} + x_{26} & −x_{16} − x_{25} & −x_{17} + x_{28} & −x_{18} − x_{27} \\ −x_{56} − x_{78} & −x_{57} + x_{68} & −x_{58} − x_{67} & +x_{37} + x_{48} & −x_{38} + x_{47} & −x_{35} − x_{46} & x_{36} − x_{45} \\ & −x_{14} + x_{23} & x_{13} + x_{24} & x_{16} + x_{25} & −x_{15} + x_{26} & x_{18} + x_{27} & −x_{17} + x_{28} \\ & +x_{58} + x_{67} & −x_{57} + x_{68} & −x_{38} + x_{47} & −x_{37} − x_{48} & +x_{36} − x_{45} & +x_{35} + x_{46} \\ & & −x_{12} + x_{34} & x_{17} + x_{28} & −x_{18} + x_{27} & −x_{15} − x_{26} & +x_{16} − x_{25} \\ & & +x_{56} + x_{78} & +x_{35} − x_{46} & +x_{36} + x_{45} & +x_{37} − x_{48} & +x_{38} + x_{47} \\ & & & x_{18} − x_{27} & x_{17} + x_{28} & −x_{16} + x_{25} & −x_{15} − x_{26} \\ & & & +x_{36} + x_{45} & −x_{35} + x_{46} & +x_{38} + x_{47} & −x_{37} + x_{48} \\ & & & & x_{12} + x_{34} & x_{13} − x_{24} & x_{14} + x_{23} \\ & & & & +x_{56} − x_{78} & +x_{57} + x_{68} & +x_{58} − x_{67} \\ & & & & & x_{14} + x_{23} & −x_{13} + x_{24} \\ & & & & & −x_{58} + x_{67} & +x_{57} + x_{68} \\ & & & & & & x_{12} + x_{34} \\ & & & & & & −x_{56} + x_{78} \end{matrix} )

and

d_{ρ^{2}}(X) = \frac{1}{2} ( \begin{matrix}−x_{12} − x_{34} & −x_{13} + x_{24} & −x_{14} − x_{23} & −x_{15} − x_{26} & −x_{16} + x_{25} & −x_{17} − x_{28} & −x_{18} + x_{27} \\ +x_{56} + x_{78} & +x_{57} − x_{68} & +x_{58} + x_{67} & −x_{37} − x_{48} & +x_{38} − x_{47} & +x_{35} + x_{46} & −x_{36} + x_{45} \\ & x_{14} + x_{23} & −x_{13} + x_{24} & −x_{16} + x_{25} & x_{15} + x_{26} & −x_{18} + x_{27} & x_{17} + x_{28} \\ & +x_{58} + x_{67} & −x_{57} + x_{68} & −x_{38} + x_{47} & −x_{37} − x_{48} & +x_{36} − x_{45} & +x_{35} + x_{46} \\ & & x_{12} + x_{34} & −x_{17} + x_{28} & x_{18} + x_{27} & x_{15} − x_{26} & −x_{16} − x_{25} \\ & & +x_{56} + x_{78} & +x_{35} − x_{46} & +x_{36} + x_{45} & +x_{37} − x_{48} & +x_{38} + x_{47} \\ & & & −x_{18} − x_{27} & −x_{17} + x_{28} & x_{16} + x_{25} & x_{15} − x_{26} \\ & & & +x_{36} + x_{45} & −x_{35} + x_{46} & +x_{38} + x_{47} & −x_{37} + x_{48} \\ & & & & −x_{12} + x_{34} & −x_{13} − x_{24} & −x_{14} + x_{23} \\ & & & & +x_{56} − x_{78} & +x_{57} + x_{68} & +x_{58} − x_{67} \\ & & & & & −x_{14} + x_{23} & x_{13} + x_{24} \\ & & & & & −x_{58} + x_{67} & +x_{57} + x_{68} \\ & & & & & & −x_{12} + x_{34} \\ & & & & & & −x_{56} + x_{78} \end{matrix} )

Since the conjugation π of ℭ is given by the matrix P = diag(1, −1, …, −1) with respect to the basis chosen above, we have

d_{π}(X) = P X P = ∑_{1<j} −x_{1j}E_{1j} + ∑_{1<i<j} x_{ij}E_{ij}.

For any skew-symmetric matrix U with entries in F, we obtain formulas for d_{α}(U) for α = ρ, ρ^{2} and π by specializing X to U. This shows that S_{3} acts faithfully on 𝔬_{8}.

Proposition 4.1 For any Cayley algebra ℭ with norm 𝔫 the action of S_{3} on the Lie algebra 𝔬(𝔫) is faithful. Further the automorphisms d_{ρ} and d_{ρ^{2}} are not inner in End_{F} (ℭ).

Proof Since ℭ isomorphic to ℭ(8) over some field extension of F, it suffices to check the claims for ℭ(8). We already know the first assertion for ℭ(8). Let det be the determinant function in End_{F} (ℭ(8)). If d_{ρ} were inner, we would have det(d_{ρ}(X)) = det(X). Let x be an indeterminate. Specializing X to x_{12} = x, x_{34} = x_{56} = x_{78} = 1 and all other entries to 0 shows that this is not the case.

□

Remark 4.2 The relations (3.4) can be explicitly verified for skew-symmetric matrices, using the formulas for d_{ρ}(X) and d_{π}(X). The general case follows by going to some field extension of F.

We next check that the action of S_{3} on 𝔬_{8} induces a faithful action of S_{3} on the Dynkin diagram of D_{4}. This gives another proof that S_{3} acts faithfully on 𝔬_{8} as outer automorphisms.

Proposition 4.3 The action defined in (3.4) induces permutations of the simple roots of 𝔬_{8}.

Proof A Cartan subalgebra ℌ of 𝔬_{8} is generated by the four diagonal blocks E_{12}, E_{34}, E_{56} and E_{78} (see [H], p. 187). The automorphism d_{ρ} of 𝔬_{8} restricts to an automorphism of ℌ. With respect to the basis u_{1} = E_{34}, u_{2} = E_{56}, u_{3} = E_{78} and u_{4} = E_{12} it is given by the orthogonal matrix

T = \frac{1}{2} (\begin{matrix}1 & 1 & 1 & −1 \\ 1 & −1 & 1 & 1 \\ 1 & −1 & −1 & −1 \\ 1 & −1 & 1 & 1 \end{matrix}) .

Let F (i) be a field extension of F such that i^{2} = −1. In view of [H], p. 188, the linear forms ε_{j}, j = 1, …, 4, in ℌ^{∗}, defined by ε_{j}(u_{k}) = iδ_{jk}, form a set of roots of 𝔬_{8} ⊗ F (i) such that the simple roots may be taken as α_{1} = ε_{1} − ε_{2}, α_{2} = ε_{2} − ε_{3}, α_{3} = ε_{3} − ε_{4} and α_{4} = ε_{3} + ε_{4} and the Dynkin diagram is

[figure: α_{1}; α_{2}]\begin{matrix}α_{3} \\ α_{4} \end{matrix}

The action induced by d_{ρ} on the dual ℌ^{∗} is given by the matrix T^{t} = T^{−1} with respect to the basis (ε_{k}). On the other hand the action of S_{3} on the Dynkin diagram permutes α_{1}, α_{3}, α_{4} and leaves α_{2} invariant. The cyclic permutation (α_{1}, α_{3}, α_{4}) extends uniquely to an automorphism of the dual ℌ^{∗}. This automorphism is given with respect to the basis (ε_{j}) by the matrix T^{t} = T^{−1}. This is the claim for the generator ρ of order 3 of S_{3}. The action of d_{π} on ℌ maps u_{4} to −u_{4} and leaves the other u’s fixed. Thus its contragredient action on ℌ^{∗} maps ε_{4} to −ε_{4} and leaves the other ε’s fixed. On the level of simple roots it permutes α_{3} and α_{4}. This concludes the proof.

□

5 Similitudes and Triality

Let ℭ be any Cayley algebra with norm 𝔫. Any proper similitude f ∈ GO^{+}(𝔫) induces an automorphism C(f) of the even Clifford algebra (C_{0}(ℭ, 𝔫), τ_{0}), which leaves the center of C_{0}(ℭ, 𝔫) invariant (see (2.5)). Thus α_{0} ∘ C(f) ∘ α_{0}^{−1} is a pair of automorphisms of (End_{F} (ℭ), σ_{𝔫}). It follows from the Skolem-Noether theorem that, for any quadratic space (V, q), automorphisms of (End_{F} (V), σ_{q}) are of the form Int(f), where f is a similitude of q. Therefore we have

α_{0} ∘ C(f) ∘ α_{0}^{−1} = (Int(f_{1}), Int(f_{2}))

for similitudes f_{1}, f_{2}. The pair (f_{1}, f_{2}) can be normalized:

Proposition 5.1 For any proper similitude f ∈ GO^{+}(𝔫) with multiplier m(f), there exist proper similitudes f_{1}, f_{2} such that

α_{0} ∘ C(f) ∘ α_{0}^{−1} = (Int(f_{1}), Int(f_{2})),

i.e., the diagram

C_{0}(ℭ, 𝔫) ⟶^{α_{0}} End_{F} (ℭ) × End_{F} (ℭ)
C(f)↓  ↓^{(Int(f_{1}), Int(f_{2}))}
C_{0}(ℭ, 𝔫) ⟶^{α_{0}} End_{F} (ℭ) × End_{F} (ℭ)

(10)

is commutative, and

(1)  m(f_{1})^{−1}f_{1}(x ⋆ y) = f(x) ⋆ f_{2}(y),
(2)  m(f)^{−1}f(x ⋆ y) = f_{2}(x) ⋆ f_{1}(y)

and

(3)  m(f_{2})^{−1}f_{2}(x ⋆ y) = f_{1}(x) ⋆ f(y).

The pair (f_{1}, f_{2}) is determined by f up to a factor (m, m^{−1}), m ∈ F^{×}, and we have

m(f_{1})m(f)m(f_{2}) = 1.

Furthermore, any of the formulas (1) to (3) implies the others.

Proof See the proof of Proposition (35.4) in [KMRT].

□

Passing from GO^{+}(𝔫) to PGO^{+}(𝔫) = GO^{+}(𝔫)/F^{×}, we get well defined automorphisms of PGO^{+}(𝔫), \overline{ρ}: [f] ↦ [f_{1}], ρ′: [f] ↦ [f_{2}], and uniqueness in (5.1) implies that ρ′ = \overline{ρ}^{2}, \overline{ρ}^{3} = 1. We thus have

(\overline{ρ}(a), \overline{ρ}^{2}(a)) = α_{0}C(a)α_{0}^{−1}  (11)

for any a ∈ P GO^{+}(𝔫).

Let \overline{π} be the automorphism of PGO^{+}(𝔫) given by \overline{π}(x) = [π]x[π], where [π] is the class in PGO(𝔫) of the conjugation map π of ℭ. It follows from (5.1) and the identity π(x ⋆ y) = π(y) ⋆ π(x) that \overline{π} \overline{ρ} = \overline{ρ}^{2}\overline{π}. Thus:

Corollary 5.2 (Global triality) The set \{\overline{π}, \overline{ρ}\} generate a subgroup of Aut_{F} (PGO^{+}(𝔫)) isomorphic to S_{3}.

Proof The fact that S_{3} acts on PGO^{+}(𝔫) follows from the relations given above. The fact that the action is faithful follows from the computations in Section 4.

□

The action of S_{3} on the Lie algebra 𝔬(𝔫) and the action on the group PGO^{+}(𝔫) are related as follows:

Lemma 5.3 Let [f] ∈ PGO^{+}(𝔫) be represented by f ∈ GO^{+}(𝔫) and \overline{ρ}([f]) (resp. \overline{ρ}^{2}([f]) be represented by f_{1} (resp. f_{2}) as above. We have, for any skew symmetric matrix U,

1) (ad ∘ C(f) ∘ ad^{−1})(U) = fUf^{−1},

2) d_{ρ}(fUf^{−1}) = f_{1}d_{ρ}(U)f_{1}^{−1}, d_{ρ}(f_{1}Uf_{1}^{−1}) = f_{2}d_{ρ}(U)f_{2}^{−1} and d_{ρ}(f_{2}Uf_{2}^{−1}) = fd_{ρ}(U)f^{−1}

3) d_{ρ^{2}}(fUf^{−1}) = f_{2}d_{ρ^{2}}(U)f_{2}^{−1}, d_{ρ^{2}}(f_{1}Uf_{1}^{−1}) = fd_{ρ^{2}}(U)f^{−1} and d_{ρ^{2}}(f_{2}Uf_{2}^{−1}) = f_{1}d_{ρ^{2}}(U)f_{1}^{−1}.

Furthermore, for the conjugation π ∈ GO(𝔫), we have

4) d_{ρ}(πUπ) = πd_{ρ^{2}}(U)π and d_{ρ^{2}}(πUπ) = πd_{ρ}(U)π.

Proof Claim 1) is already in (2.5). We check the first formula of 2). By definition we have α_{0} ∘ ad^{−1}(U) = (d_{ρ}(U), d_{ρ^{2}}(U)). Thus

(d_{ρ}(fUf^{−1}), d_{ρ^{2}}(fUf^{−1}) = α_{0} ∘ ad^{−1}(fUf^{−1})
= (α_{0} ∘ C(f) ∘ ad^{−1})(U)
= (α_{0} ∘ C(f) ∘ α_{0}^{−1}) ∘ (α_{0} ∘ ad^{−1})(U)
= (α_{0} ∘ C(f) ∘ α_{0}^{−1})(d_{ρ}(U), d_{ρ^{2}}(U))
= (f_{1}d_{ρ}(U)f_{1}^{−1}), f_{2}d_{ρ^{2}}(U)f_{2}^{−1})

by (5.1). The proofs of the other formulas in 2) and 3) are similar. Claim 4) is the relation d_{π}d_{ρ} = d_{ρ^{2}}d_{π} in (3.4).

□

Since S_{3} acts on PGO^{+}(𝔫) we may form the semidirect product PGO^{+}(𝔫)⋊S_{3}. The group PGO^{+}(𝔫) ⋊ S_{3} consists of pairs (b, β), where b ∈ PGO^{+}(𝔫) and β ∈ S_{3} with the product (b, β) · (b′, β′) = (bβ(b′), ββ′). For any β ∈ S_{3} we denote the corresponding automorphism of 𝔬(𝔫) by d_{β}.

Proposition 5.4 1) The group PGO^{+}(𝔫) acts on 𝔬(𝔫) through inner automorphisms and, for any b ∈ PGO^{+}(𝔫), β ∈ S_{3}, s ∈ 𝔬(𝔫), we have β(b)d_{β}(s) = d_{β}(bs).

2) The group PGO^{+}(𝔫) ⋊ S_{3} acts on 𝔬(𝔫) through the formula (b, β)(s) = bd_{β}(s) and Aut_{F} (𝔬(𝔫)) = PGO^{+}(𝔫) ⋊ S_{3}.

Proof Claim 1) and the first part of 2) follow from (5.3). The last claim of 2) is for example in [J].

□

Proposition 5.5 The differential of the action of S_{3} on PGO^{+}(𝔫) in (5.2) is the action of S_{3} defined on 𝔬(𝔫).

Proof We first recall the definition of the differential. Denote by F [ε] the F -algebra of dual numbers, i.e., F [ε] = F · 1 ⊕ F · ε with multiplication given by ε^{2} = 0. Let κ: F [ε] → F, ε ↦ 0, be the augmentation map. Let G be an algebraic group. The kernel of the induced map G(F [ε]) ⟶^{G(κ)} G(F) is the Lie algebra Lie(G) of G. If G ⊂ GL_{n}(F), Lie(G) = {a ∈ M_{n}(F) | 1 + aε ∈ G(F [ε])}. Any homomorphism of algebraic groups f: G → H induces a commutative diagram

G(F [ε]) ⟶^{f_{F [ε]}} H(F [ε])
G(κ)↓  ↓^{H(κ)}
G(F) ⟶^{f_{F}} H(F)

and hence defines an F -linear map df: Lie(G) → Lie(H) which is the differential of f.

We only check that the differential of \overline{ρ} in (11) is d_{ρ} as defined in (3.4). Let [g] = [1 + aε] ∈ PGO^{+}(𝔫 ⊗ F [ε]), so that [a] ∈ Lie(PGO^{+}(𝔫)) = Lie(GO^{+}(𝔫))/F. By definition of the differential we have [\overline{ρ}(g)] = [1 + d\overline{ρ}(a)ε] and by definition of triality

m(\overline{ρ}(g)^{−1})\overline{ρ}(g)(x ⋆ y) = g(x) ⋆ \overline{ρ}^{2}(g)(y).

Modulo scalars we have

(1 + d\overline{ρ}(a)ε)(x ⋆ y) = (1 + aε)(x) ⋆ (1 + d\overline{ρ}^{2}(a)ε)(y)

or

d\overline{ρ}(a)(x ⋆ y) = a(x) ⋆ y + x ⋆ d\overline{ρ}^{2}(a)(y)

hence d\overline{ρ} = d_{ρ} by (3.2) (local triality).

□

6 Orthogonal Involutions

Let GO_{n}(F) = {a ∈ GL_{n}(F) | aa^{t} ∈ F^{×}} denote the group of similitudes of the n-dimensional identity quadratic form < 1, …, 1 >. The element m(a) = aa^{t} is the multiplier of the similitude a. Elements of GO_{n}(F) act on (M_{n}(F), t) through inner automorphisms, a(x) = axa^{−1}, a ∈ GO_{n}(F), and

Aut_{F} (M_{n}(F), t) = GO_{n}(F)/F^{×} = PGO_{n}(F).  (12)

We assume that n = 2l is even. Proper similitudes a satisfy det(a) = m(a)^{l} and form a subgroup GO_{2l}^{+}(F) of GO_{2l}(F). Going modulo the center we get the group of classes of similitudes PGO_{2l}(F) and the group PGO_{2l}^{+}(F) of classes of proper similitudes.

The group PGO_{2l}^{+}(F) can also be described as a group of automorphisms; let

Skew_{2l}(F) = {X ∈ M_{2l}(F) | X^{t} = −X}

be the set of skew-symmetric (2l × 2l)-matrices. The pfaffian

pf : Skew_{2l}(F) → F

is a homogeneous polynomial function of degree l such that pf(X)^{2} = det(X); further, pf(AXA^{t}) = det(A) pf(X) for A ∈ M_{2l}(F) and X ∈ Skew_{2l}(F). For [A] ∈ PGO_{2l}(F) we have pf(AXA^{−1}) = pf(X) if and only if and [A] ∈ PGO_{2l}^{+}(F). Thus

Aut_{F} (M_{2l}(F), t, pf) = GO_{2l}^{+}(F)/F^{×} = PGO_{2l}^{+}(F).  (13)

Let F̃ be a separable closure of F and let Γ = Gal(F̃/F) be the absolute Galois group of F. Let H^{1}(F, G) = H^{1}(Γ, G(F̃)) be the pointed set of Galois cohomology of Γ with values in G. The set H^{1}(F, G) classifies G-torsors (principal homogeneous spaces for G over F). Assume that G is the automorphism group of some tensor w over F; then H^{1}(F, G) classifies isomorphism classes of F̃/F -forms of w. Let (A, σ) be a central simple algebra of degree n over F with an F -linear involution σ. We say that σ is of orthogonal type if we have an isomorphism (A, σ) ⊗ F̃ ≃ (M_{n}(F̃), t) over a separable closure F̃ of F. We call such an isomorphism a standard splitting of (A, σ). In view of (12) the pointed set H^{1}(F, PGO_{n}) classifies central simple F -algebras (A, σ) of degree n with orthogonal involutions, the point being the class of (M_{n}(F), t).

We give a similar description of H^{1}(F, PGO_{2l}^{+}). For any algebra A with involution σ, let

Skew(A, σ) = {x ∈ A | σ(x) = −x}

be the set of skew elements.

Proposition 6.1 The pointed set H^{1}(F, PGO_{2l}^{+}) classifies triples (A, σ, Pfrd_{A}), where A is central simple over F of degree 2l, σ is an involution of orthogonal type on A and Pfrd_{A} is a polynomial function of degree l on Skew(A, σ) such that Pfrd_{A}(x)^{2} = Nrd_{A}(x) for x ∈ Skew(A, σ). The point is the class of the triple (M_{2l}(F), t, pf). Further, the map Pfrd_{A} satisfies Pfrd_{A}(axσ(a)) = Nrd_{A}(a) Pfrd_{A}(x) for x ∈ Skew(A, σ) and a ∈ A.

Proof Let (a_{s})_{s∈Γ}, a_{s} ∈ PGO_{2l}^{+}(F̃), be a cocycle with values in PGO_{2l}^{+}. Its image as a cocycle with values in PGO_{2l} defines the central simple algebra

A = {x ∈ M_{2l}(F̃) | a_{s}s(x)a_{s}^{−1} = x for all s ∈ Γ}

with an orthogonal involution σ which is the restriction of the transpose t. We claim that the pfaffian on Skew_{2l}(F̃) restricts to a function Pfrd_{A}: Skew(A, σ) → F. It suffices to check that s(pf(x)) = pf(x) for x ∈ Skew(A, σ) and s ∈ Γ. We have

pf(x) = pf(a_{s}s(x)a_{s}^{−1}) = pf(a_{s}s(x)a_{s}^{t}m(a_{s})^{−1})
= pf(s(x)) det(a_{s})m(a_{s})^{−l} = pf(s(x)) (14)

since det(a_{s}) = m(a_{s})^{l} for a_{s} ∈ PGO_{2l}^{+}(F̃). The claim then follows from pf(s(x)) = s(pf(x)). The given properties of the polynomial map Pfrd_{A} follow from the corresponding properties of the pfaffian.

□

We call a map Pfrd_{A} as in (6.1) a reduced pfaffian. We have an exact sequence of Γ-groups

1 ⟶ PGO_{2l}^{+}(F̃) ⟶^{i} PGO_{2l}(F̃) ⟶^{d} μ_{2} ⟶ 1

where, for a similitude a with multiplier m(a), d([a]) = m(a)^{−l} det(a) ∈ μ_{2}. The sequence induces an exact sequence of pointed sets in Galois cohomology:

… ⟶ μ_{2} ⟶ H^{1}(F, PGO_{2l}^{+}) ⟶^{i_{∗}} H^{1}(F, PGO_{2l}) ⟶^{d_{∗}} H^{1}(F, μ_{2})

The image under i_{∗} of the class of the triple (A, σ, Pfrd_{A}) is the class of the pair (A, σ). We describe the map d_{∗}:

Proposition 6.2 If H^{1}(F, μ_{2}) is identified with F^{×}/F^{×2}, then d_{∗}([A, σ]) is the square class [Nrd_{A}(x)] ∈ F^{×}/F^{×2} of any skew-symmetric unit of A.

Proof We recall the identification of H^{1}(F, μ_{2}) with F^{×}/F^{×2}: any cocycle (b_{s})_{s∈Γ} ∈ Z^{1}(F, μ_{2}) can be written as b_{s} = s(\sqrt{b})\sqrt{b}^{−1} for b ∈ F^{×} and \sqrt{b} ∈ F̃^{×}. Then the square class [b] ∈ F^{×}/F^{×2} corresponds to the cocycle (b_{s}). Let (a_{s})_{s∈Γ}, a_{s} ∈ PGO_{2l}(F̃), be a cocycle with values in PGO_{2l}. By (14) we have d_{∗}([a_{s}]) = m(a_{s})^{−l} det(a_{s}) = s(pf(x)) pf(x)^{−1} for any skew-symmetric unit x ∈ A. Thus d_{∗}([A, σ]) = [pf(x)^{2}] = [Nrd_{A}(x)], as claimed.

□

Remark 6.3 The square class [Nrd_{A}(x)] ∈ F^{×}/F^{×2} which, in view of (6.2), is independent of the choice of the skew-symmetric unit x, is the discriminant disc(σ) of the involution σ (see [KPS]). The image of the map s_{∗} in H^{1}(F, PGO_{2l}) consists of classes [(A, σ)] such that σ has trivial discriminant. Given a class [(A, σ)] in H^{1}(F, PGO_{2l}) with disc(σ) = 1, there could be two possible choices of lifts in H^{1}(F, PGO_{2l}^{+}), namely [(A, σ, ± Pfrd_{A})].

We briefly recall how the discriminant is related with the Clifford algebra and refer to [KMRT] for details. The map

C: PGO_{2l}(F̃) → AutF̃(C_{0}(2l), τ_{0})

defined in (2) induces a map in Galois cohomology. The image C(ξ) of a cocycle ξ corresponding to an algebra with orthogonal involution (A, σ) defines an algebra C(A, σ) with an involution \underline{σ}. The algebra C(A, σ) is called the Clifford algebra of (A, σ). Any isomorphism ϕ: (A, σ_{A}) ⟶^{∼} (A′, σ_{A′}) of algebras with orthogonal involutions induces an isomorphism C(ϕ) of the corresponding Clifford algebras. The algebra C(A, σ) has a similar structure as the even Clifford algebra of a quadratic space of even dimension (see (2.1)); its center Z(A, σ) is quadratic étale over F and C(A, σ) is central separable over Z(A, σ). If (A, σ) = (End_{F} (V), σ_{q}), then (C(A, σ), \underline{σ}) is canonically isomorphic to (C_{0}(V, q), τ_{0}).

There is an embedding

η_{A}: Skew(A, σ) → Skew(C(A, σ), \underline{σ})  (15)

which is the map

ad^{−1} : 𝔬(q) → [V, V] ⊂ C_{0}(V, q)

if (A, σ) = (End_{F} (V), σ_{q}).

The center Z(A, σ) of C(A, σ) has a generator z such that z^{2} = d ∈ F^{×}. If (A, σ) = (End_{F} (V), σ_{q}), we have z^{2} = (−1)^{l} det(b_{q}) where det(b_{q}) is the determinant of the matrix b of b_{q} with respect to some basis of V. The square class disc(q) = [det(b_{q})] ∈ F^{×}/F^{×2} = H^{1}(F, μ_{2}), is the (unsigned) discriminant of q. The isomorphism class of the center of C_{0}(V, q) is determined uniquely by disc(q). The involution σ_{q} is given by x ↦ bx^{t}b^{−1} for x ∈ M_{2l}(F) and it follows from (6.2) that disc(σ_{q}) = [det(b)] = disc(q)]. For (A, σ) arbitrary we similarly write z^{2} = (−1)^{l}d and the square class [d] ∈ F^{×}/F^{×2} of d is the discriminant disc(σ) of σ as defined in (6.3) (see [KMRT, (8.25)]^{1}).

1Our definition of the discriminant of an orthogonal involution is not consistent with the definition given in [KMRT]. There it is called the determinant and the discriminant in [KMRT] is the class of (−1)^{l} · d modulo squares.

From now on we assume that A is of degree a multiple of 4, so that l as above is even. The center Z(A, σ) is generated by an element z such that z^{2} = d with [d] = disc(σ). If disc(σ) = 1, then Z(A, σ) ≃ F × F. The choice of an isomorphism γ: Z(A, σ) ⟶^{∼} F × F corresponds to a choice of a sign of Pfrd_{A}. Thus:

Corollary 6.4 Let n be a multiple of 4. The pointed set H^{1}(F, PGO_{n}^{+}) classifies triples (A, σ, γ) where γ is an F -algebra isomorphism γ: Z(A, σ) ⟶^{∼} F × F.

7 Trialitarian algebras

Let (A, σ_{A}) be a central simple algebra of degree 8 over F, with an orthogonal involution σ_{A} of discriminant 1. With the notation of (6.4), the choice of an isomorphism γ: Z(A, σ_{A}) ⟶^{∼} F × F leads to a decomposition

(C(A, σ_{A}), \underline{σ}) ≃ (B, σ_{B}) × (C, σ_{C})

The algebras B and C are of degree 8 and the involutions σ_{B}, σ_{C} are of orthogonal type (see (2.1)). Thus, given a cocycle ξ ∈ H^{1}(F, PGO_{8}^{+}), there exist algebras of degree 8 ((A, σ_{A}), (B, σ_{B}), (C, σ_{C})) with orthogonal involutions together with an isomorphism

α_{A}: (C(A, σ_{A}), \underline{σ}) ≃ (B, σ_{B}) × (C, σ_{C}).  (16)

If (A, σ) = (M_{8}(F), t), then (B, σ_{B}) ≃ (M_{8}(F), t) ≃ (C, σ_{C}) and formula (4) gives an explicit isomorphism α_{M_{8}(F)} = α_{0}. The triple (M_{8}(F), t, α_{0}) corresponds to the point of H^{1}(F, PGO_{8}^{+}).

Lemma 7.1 The trialitarian action on PGO_{8}^{+}(F̃) is compatible with the action of the Galois group Γ = Gal(F̃/F) and leads to an action of S_{3} on H^{1}(F, PGO_{8}^{+}).

Proof Let a ∈ PGO_{8}^{+}(F̃) = AutF̃(M_{8}(F̃), t, pf). The action of Γ on the group PGO_{8}^{+}(F̃) is given by s(a) = sas^{−1} for s ∈ Γ. The trialitarian action is defined as (\overline{ρ}(a), \overline{ρ}^{2}(a)) = (α_{0} ⊗ 1F̃) ∘ C(a) ∘ (α_{0} ⊗ 1F̃)^{−1} (see (11)). Since α_{0} is defined over F and the functor C commutes with the action of Γ, we have s(\overline{ρ}(a)) = \overline{ρ}(s(a)). One can also check that s\overline{π} = \overline{π}s.

□

Our next aim is to describe this action on H^{1}(F, PGO_{8}^{+}), in terms of the classified objects. Let β_{A}: A ⊗ F̃ ⟶^{∼} M_{8}(F̃) be a standard splitting of (A, σ_{A}, Pfrd_{A}) and let a_{s} = β_{A} ∘ (1 ⊗ s) ∘ β_{A}^{−1} ∘ s^{−1} ∈ PGO_{8}^{+}(F̃) be the corresponding cocycle. Let

\begin{aligned}B & = & \{y ∈ M_{8}(F̃) | \overline{ρ}(a_{s})(sy) = y, s ∈ Γ\} and \\ C & = & \{z ∈ M_{8}(F̃) | \overline{ρ}^{2}(a_{s})(sz) = z, s ∈ Γ\} \end{aligned}

be the central simple algebras defined by the cocycles \overline{ρ}(a_{s}), resp. \overline{ρ}^{2}(a_{s}). The involutions σ_{B}, resp. σ_{C} are the restrictions of the transpose to B, resp. C and the pfaffians Pfrd_{B} and Pfrd_{C} are the restrictions of the usual pfaffian pf. The two splittings β_{B}: B ⊗ F̃ ⟶^{∼} M_{8}(F̃), resp. β_{C}: C ⊗ F̃ ⟶^{∼} M_{8}(F̃) are given by multiplication in M_{8}(F̃). The Clifford algebra of (A, σ) is

C(A, σ) = {u ∈ C_{0}(8) ⊗ F̃ | C(a_{s})(su) = u, s ∈ Γ}.

Lemma 7.2 The restriction of α_{0} ⊗ 1F̃: C_{0}(8) ⊗ F̃ ⟶^{∼} M_{8}(F̃) × M_{8}(F̃) to C(A, σ) induces an isomorphism α_{A}: (C(A, σ_{A}), \underline{σ}) ⟶^{∼} (B, σ_{B}) × (C, σ_{C}) such that the diagram

C(A, σ_{A}) ⊗ F̃ ⟶^{α_{A}⊗1F̃} B ⊗ F̃ × C ⊗ F̃
C(β_{A})↓  ↓^{(β_{B}, β_{C})}
C_{0}(8) ⊗ F̃ ⟶^{α_{0}⊗1F̃} M_{8}(F̃) × M_{8}(F̃)

commutes.

Proof Let (y, z) = α_{0}(x) for x ∈ C(A, σ_{A}). We have to check that (\overline{ρ}(a_{s}), \overline{ρ}^{2}(a_{s}))(sy, sz) = (y, z) for all s ∈ Γ.

Denote α_{0} ⊗ 1F̃ = α_{0}; we have, since α_{0}s = sα_{0} and C(a_{s})sx = x,

(\overline{ρ}(a_{s}), \overline{ρ}^{2}(a_{s}))(sy, sz) = α_{0}C(a_{s})α_{0}^{−1}(sy, sz)
= α_{0}C(a_{s})α_{0}^{−1}s(α_{0}(x))
= α_{0}C(a_{s})sx = α_{0}(x) = (y, z).
□

Repeating the same procedure for (B, σ_{B}), resp. (C, σ_{C}) in (7.2) we get isomorphisms

\begin{aligned}α_{B}: (C(B, σ_{B}), \underline{σ}) & ⟶^{∼} & (C, σ_{C}) × (A, σ_{A}) \\ α_{C}: (C(C, σ_{C}), \underline{σ}) & ⟶^{∼} & (A, σ_{A}) × (B, σ_{B}). \end{aligned}  (17)

In particular the \overline{ρ}-action on H^{1}(F, PGO_{8}^{+}) corresponds to the permutation (A, B, C) ↦ (B, C, A). On can similarly check that \overline{π} switches B and C.

The split exact sequence

1 ⟶ PGO_{8}^{+}(F̃) ⟶ PGO_{8}^{+}(F̃) ⋊ S_{3} ⟶ S_{3} ⟶ 1

induces an exact sequence of pointed sets in Galois cohomology

⟶ H^{1}(F, PGO_{8}^{+}) ⟶ H^{1}(F, PGO_{8}^{+} ⋊S_{3}) ⟶ H^{1}(F, S_{3}).  (18)

The set H^{1}(F, S_{3}) classifies cubic étale F -algebras L, i.e., F -algebras L with L ⊗ F̃ ≃ F̃ × F̃ × F̃ (see [KMRT]). The set H^{1}(F, PGO_{8}^{+}) classifies triples (A, B, C) as in (7.2). Following [KMRT], we introduce algebraic objects which are classified by H^{1}(F, PGO_{8}^{+} ⋊S_{3}).

The triple (A, B, C) gives rise to an F ×F ×F -algebra T = A×B ×C with an involution σ_{T} = (σ_{A}, σ_{B}, σ_{C}). The triple (α_{A}, α_{B}, α_{C}) is an F × F × F -isomorphism of C(T, σ_{T}) with the (F × F × F) × (F × F)-algebra (B × C) × (C × A) × (A × B); we write the cubic étale algebra L = F × F × F as a column (F, F, F)^{t} and its discriminant Δ = F × F as a row (F, F), so that L ⊗ Δ is represented by (3 × 2)-matrices: L ⊗ Δ = (\begin{matrix}F & F \\ F & F \\ F & F \end{matrix} ) and (α_{A}, α_{B}, α_{C})

has values in (\begin{matrix}B & C \\ C & A \\ A & B \end{matrix}). Let (α_{A}^{1}, α_{A}^{2}), resp. (α_{B}^{1}, α_{B}^{2}), (α_{C}^{1}, α_{C}^{2}) be the two components of α_{A},

resp. α_{B}, α_{C}. We have T ⊗ Δ = (\begin{matrix}A \\ B \\ C \end{matrix}) ⊗ (F × F) = (\begin{matrix}A & A \\ B & B \\ C & C \end{matrix}); let

α_{T} = (\begin{matrix}α_{C}^{1} & α_{B}^{2} \\ α_{A}^{1} & α_{C}^{2} \\ α_{B}^{1} & α_{A}^{2} \end{matrix}): C(T, σ_{T}) → T ⊗ Δ.  (19)

The map α_{T} is F -linear, but not L-linear. To restore L-linearity we have to twist the action of L⊗Δ = (\begin{matrix}F & F \\ F & F \\ F & F \end{matrix} ) on (\begin{matrix}A & A \\ B & B \\ C & C \end{matrix}) through an automorphism of L⊗Δ; in fact L⊗Δ is a S_{3}-Galois F -algebra with the action of S_{3} given as

ρ̃(\begin{matrix}a_{1} & a_{2} \\ b_{1} & b_{2} \\ c_{1} & c_{2} \end{matrix}) = (\begin{matrix}c_{1} & b_{2} \\ a_{1} & c_{2} \\ b_{1} & a_{2} \end{matrix}) and π̃(\begin{matrix}a_{1} & a_{2} \\ b_{1} & b_{2} \\ c_{1} & c_{2} \end{matrix}) = (\begin{matrix}a_{2} & a_{1} \\ b_{2} & b_{1} \\ c_{2} & c_{1} \end{matrix}).  (20)

For any L ⊗ Δ-module V with L and Δ as above, let ρ̃V be the F -module V with the action of L ⊗ Δ twisted through ρ̃, i.e., ξ · x = ρ̃(ξ)x for ξ ∈ L ⊗ Δ and x ∈ T. For T = A × B × C and Δ = F × F, the map (19) viewed as a map

α_{T} = (\begin{matrix}α_{C}^{1} & α_{B}^{2} \\ α_{A}^{1} & α_{C}^{2} \\ α_{B}^{1} & α_{A}^{2} \end{matrix}): C(T, σ_{T}) → ρ̃(T ⊗ Δ)  (21)

is L-linear. For (A, σ) = (M_{8}(F), t), we have (B, σ) = (C, σ) = (M_{8}(F), t), T = M_{8}(F) × M_{8}(F) × M_{8}(F) and (α_{A}, α_{B}, α_{C}) = (α_{0}, α_{0}, α_{0}). We denote the corresponding isomorphism α_{T} by α̃; it is uniquely determined by:

α̃ ∘ ad^{−1} (\begin{matrix}x \\ y \\ z \end{matrix}) = (\begin{matrix}d_{ρ}(z) & d_{ρ^{2}}(y) \\ d_{ρ}(x) & d_{ρ^{2}}(z) \\ d_{ρ}(y) & d_{ρ^{2}}(x) \end{matrix})  (22)

for (x, y, z) ∈ 𝔬(8) × 𝔬(8) × 𝔬(8). Let

M(F) = M_{8}(F) × M_{8}(F) × M_{8}(F) = M_{8}(F × F × F)

viewed a a central separable algebra over F × F × F. Let t be the involution on M(F)which is transpose on each factor. We call the data

(M(F), F × F × F, t, α̃)

the standard trialitarian algebra over F. We also abbreviate the data by M(F).

Let ϕ be an F -automorphism of (M(F), t); ϕ restricts to an automorphism γ of the center F × F × F, i.e., ϕ is γ-semilinear. We claim that ϕ extends to a γ-semilinear automorphism C(ϕ) of the Clifford algebra C(M(F), t). More generally we have:

Lemma 7.3 Let L be an étale F -algebra and let (V, q) be a quadratic space over L.

1) Let γ be an F -automorphism of L and let f be a γ-semilinear similitude of (V, q), i.e. q(fv) = m(f)γ(q(v)) for v ∈ V, m(f) ∈ L^{×}. Then f extends to a γ-semilinear automorphism of C_{0}(V, q).

2) Any F -automorphism ϕ of (End_{L}(V), σ_{q}) restricts to an automorphism γ of the center L and is of the form ϕ(x) = fxf^{−1} for some γ-semilinear similitude f of V. Further it induces a γ-semilinear automorphism C(ϕ) of C(End_{L}(V), σ_{q})) = C_{0}(V, q).

Proof The proof of 1) is similar to the proof of (2.5) (see also [W]). We prove claim 2); the automorphism ϕ induces an automorphism of End_{F} (V), hence is of the form Int(f) with f ∈ GL(V). The fact that ϕ commutes with σq implies that f is a semilinear similitude.

□

Let ϕ be an F -algebra automorphism of M(F). Its restriction γ to the center F × F × F of M(F) induces an automorphism Δ(ϕ) of the discriminant Δ = F × F. We say that ϕ is an automorphism of the standard trialitarian algebra M(F) if α̃ ∘ C(ϕ) = (ϕ ⊗ Δ(ϕ)) ∘ α̃, i.e., the following diagram is commutative:

\begin{matrix}C(M(F), t) & \underrightarrow{α̃} & ρ̃(M(F) ⊗ Δ) \\ ^{C(ϕ)}↓ & & ↓^{ϕ⊗Δ(ϕ)} \\ C(M(F), t) & \underrightarrow{α̃} & ρ̃(M(F) ⊗ Δ) \end{matrix}  (23)

Proposition 7.4 We have Aut_{F} (M(F)) ≃ PGO_{8}^{+}(F) ⋊ S_{3}.

Proof Since any automorphism of M(F) restricts to an automorphism of the center F × F × F of M(F), there exists a homomorphism θ: Aut_{F} (M(F)) → Aut_{F} (F × F × F) = S_{3}. We show that θ is surjective by exhibiting a section s. Let ρ_{0} be the 3-cycle (1 2 3), so that ρ_{0} acts on F × F × F as ρ_{0}(x, y, z) = (z, x, y), and let π_{0} the transposition (2 3) ∈ S_{3}, so that π_{0}(x, y, z) = (x, z, y). We define for U, V, and W ∈ M_{8}(F): s(ρ_{0})(U, V, W) = (W, U, V) and s(π_{0})(U, V, W) = (P UP, P W P, P V P)), where P is the matrix of the Cayley conjugation π with respect to an orthogonal basis as in Section 5. Since s(ρ_{0})s(π_{0}) = s(π_{0})s(ρ_{0}^{2}), s is a section. An automorphism ϕ belonging to the kernel of θ restricts to identity on the center of M(F); it is (F × F × F)-linear, hence of the form (a, b, c), for a, b and c ∈ PGO_{8}^{+}(F). Further it induces the identity on Δ. It then follows from the commutativity of (23) that

C_{0}(8) ⟶^{α_{0}} M_{8}(F) × M_{8}(F)
C(a)↓  ↓^{(b, c)}
C_{0}(8) ⟶^{α_{0}} M_{8}(F) × M_{8}(F)

commutes. By (11) we have that b = \overline{ρ}(a), c = \overline{ρ}^{2}(a) and the map ϕ ↦ a gives an isomorphism of the kernel of θ with PGO_{8}^{+}(F). Thus Aut_{F} (M(F)) ≃ PGO_{8}^{+} ⋊S_{3}, as claimed.

□

Remark 7.5 An explicit describtion of the action of the semidirect product PGO_{8}^{+}(F) ⋊ S_{3} follows from the proof of (7.4): let (b, β) ∈ PGO_{8}^{+}(F) ⋊ S_{3} and let (U, V, W) ∈ M(F) = M_{8}(F)×M_{8}(F)×M_{8}(F). Let b = [B], \overline{ρ}(b) = [B_{1}] and \overline{ρ}^{2}(b) = [B_{2}] for B, B_{1}, B_{2} ∈ GO_{8}^{+}(F). We have

\begin{aligned}(b, 1)(U, V, W) & = & (BUB^{−1}, B_{1}V B_{1}^{−1}, B_{2}W B_{2}^{−1}), \\ (1, ρ_{0})(U, V, W) & = & (W, U, V) \\ (1, π_{0})(U, V, W) & = & (P UP, P W P, P V P) \end{aligned}

for ρ_{0}, π_{0} and P as above.

Let L be a cubic étale F-algebra. To L is associated functionally the étale quadratic algebra Δ = Δ(L) which is the discriminant algebra of L ([KMRT, p-290)] Let sgn : S_{3} → μ_{2} be the signature map and let d : H^{1}(F, S_{3}) → H^{1}(F, μ_{2}) ≃ F^{⋆}/F^{⋆^{2}} be the induced map. Let d(L) = [D], D ∈ F^{⋆}. Then, Δ ≃ F [x]/(x^{2} − D). The algebra L is cyclic if and only if Δ ≃ F × F. For any cyclic F-algebra L, there exists on F-algebra automorphism ρ of L of order 3. The algebra L ⊗ Δ over Δ is cyclic and let ρ be a generator of L ⊗ Δ over Δ. For any L module V, we denote by ^{ρ}(V ⊗ Δ) the module V ⊗ Δ with L ⊗ Δ-action twisted through ρ.

Let T be a central separable L-algebra, with an orthogonal involution σ_{T}. We say that T is pre-trialitarian if there exists an L-isomorphism

α_{T} : (C(T, σ_{T}), \underline{σ}) ⟶^{∼} ^{ρ}((T, σ_{T}) ⊗_{F} Δ).  (24)

The standard trialitarian algebra T̃ = M(F) is pre-trialitarian with α_{T} = α̃.

An isomorphism of pre-trialitarian algebras Ψ : (T, L, σ_{T} , α_{T}) ⟶^{∼} (T′, L′, σ_{T′}, α_{T′}) is a pair (ψ : T ⟶^{∼} T′, ϕ : L ⟶^{∼} L′) of F -isomorphisms where ψ is ϕ-semilinear, such that the diagram

\begin{matrix}C(T, σ_{T}) & ⟶^{α_{T}} & ^{ρ}(T ⊗ Δ(L)) \\ ^{C(ψ)}↓ & & ↓^{ψ⊗Δ(ϕ)} \\ C(′T, σ_{T′}) & ⟶^{α_{T′}} & ^{ρ}(T′ ⊗ Δ(L′)) \end{matrix}  (25)

commutes. For a field extension F′/F and any pre-trialitarian F -algebra T, T′ = T ⊗_{F} F′ is pre-trialitarian over F′ in an obvious way. We say that a pre-trialitarian F -algebra T is trialitarian over F if there exists a field extension F′/F and an isomorphism T ⊗_{F} F′ ⟶^{∼} M(F′), where M(F′) is the standard trialitarian algebra over F′.

The algebra A × B × C with A, B and C as in (7.2) is trialitarian over F, with L = F × F × F and α_{T} as defined in (21):

Proposition 7.6 Trialitarian algebras over a field F are classified by the pointed set H^{1}(F, PGO_{8}^{+} ⋊S_{3}), with the class of the standard trialitarian algebra M(F) as the point. The map H^{1}(F, PGO_{8}^{+}) → H^{1}(F, PGO_{8}^{+} ⋊S_{3}) of (18) associates to a central simple F - algebra (A, σ_{A}) of degree 8, with orthogonal involution σ_{A} with discriminant 1, the data (A × B × C, F × F × F, (σ_{A}, σ_{B}, σ_{C}), (α_{A}, α_{B}, α_{C})) as in (7.2).

Let T = (T, L, σ_{T} , α_{T}) be a trialitarian algebra and let

η_{T} : Skew(T, σ_{T}) → Skew(C(T, σ_{T}), \underline{σ})

be the canonical embedding (15). Let Z_{T} be the center of C(T, σ_{T}). The isomorphism α_{T} : C(T, σ_{T}) ⟶^{∼} ^{ρ}(T ⊗ Δ) restricts to an F -isomorphism α′: Z_{T} ⟶^{∼} L ⊗ Δ and we have an isomorphism 1⊗α′^{−1} of Skew(T, σ_{T})⊗_{F} Δ = Skew(T, σ_{T})⊗_{L}(L⊗Δ) with Skew(T, σ_{T})⊗_{L}Z_{T} . In view of (2.4) we have Skew(C(T, σ_{T}), \underline{σ}) = η_{T} (Skew(T, σ_{T}))⊗_{L} Z_{T} so that α_{T} ∘(η_{T} ⊗1_{Z_{T}}) is an isomorphism Skew(T, σ_{T}) ⊗_{L} Z_{T} ⟶^{∼} ^{ρ}(Skew(T, σ_{T}) ⊗_{F} Δ). The composition of these two isomorphisms is an isomorphism

α_{ρ}: Skew(T, σ_{T}) ⊗_{F} Δ ⟶^{∼} ^{ρ}(Skew(T, σ_{T}) ⊗_{F} Δ)

which, in turn, can be viewed as a ρ-semilinear automorphism of the (L ⊗ Δ)-Lie algebra Skew((T, σ_{T})⊗_{F} Δ). Let α_{π} = 1_{Skew(T,σ_{T})} ⊗ι, where ι is conjugation on the quadratic algebra Δ.

Proposition 7.7 The pair of automorphisms α_{ρ} and α_{π} constitutes a descent datum on the Lie algebra Skew(T, σ_{T}) ⊗_{F} Δ from L ⊗ Δ to F with respect to the Galois group S_{3}. The fixed points of the S_{3}-action

𝔬(T) = {x ∈ Skew(T, σ_{T}) ⊗_{F} Δ | α_{γ}(x) = x for all γ ∈ S_{3}}

is a Lie algebra (of type D_{4}) associated with the triality T. For the standard trialitarian algebra T = M(F), 𝔬(T) = 𝔬_{8}.

Proof The fact that α_{ρ} and α_{π} generate a S_{3}-semilinear action needs to be checked only in the standard case, by going over to a standard splitting. For the standard trialitarian algebra T = M_{8}(F) × M_{8}(F) × M_{8}(F) we fix an isomorphism Δ = F × F and, by (20), for x_{i}, y_{i},z_{i} ∈ 𝔬_{8}(F)

α_{ρ}(\begin{matrix}x_{1} & x_{2} \\ y_{1} & y_{2} \\ z_{1} & z_{2} \end{matrix}) = (\begin{matrix}d_{ρ}(z_{1}) & d_{ρ^{2}}(y_{2}) \\ d_{ρ}(x_{1}) & d_{ρ^{2}}(z_{2}) \\ d_{ρ}(y_{1}) & d_{ρ^{2}}(x_{2}) \end{matrix}) and α_{π} (\begin{matrix}x_{1} & x_{2} \\ y_{1} & y_{2} \\ z_{1} & z_{2} \end{matrix}) = (\begin{matrix}x_{2} & x_{1} \\ y_{2} & y_{1} \\ z_{2} & z_{1} \end{matrix}).

The first claim then follows from this explicit description of α_{ρ} and α_{π}. We leave the last claim as an exercise.

□

Remark 7.8 If L/F is cubic cyclic the discriminant Δ is split, ^{ρ}(L ⊗ Δ) = ^{ρ}L × ^{ρ^{2}}L and we have

α_{T} : (C(T, σ_{T}), \underline{σ}) ⟶^{∼} ^{ρ}(T, σ) × ^{ρ^{2}}(T, σ).

We say in this case that T is cyclic trialitarian. Cyclic trialitarian algebras are classified by the pointed set H^{1}(F, PGO_{8}^{+} ⋊A_{3}). The restriction of α_{T} to Skew(T, σ_{T}) is (α_{ρ}, α_{ρ^{2}}) and we get a Galois descent datum for Skew(T, σ_{T}) from L to F.

8 Generic torsors

Let H be a closed subgroup of GL_{n}. We have an exact sequence of pointed sets

1 → H(F) → GL_{n}(F) → (GL_{n} /H)(F) ⟶^{δ}H^{1}(F, H) → H^{1}(F, GL_{n})  (26)

(see for example [KMRT]) and H^{1}(F, GL_{n}) = 0 by Hilbert 90. We recall that the set H^{1}(F, H) classifies H-torsors. Let F (GL_{n} /H) be the function field of the homogeneous variety (GL_{n} /H) which is defined over F. Let η be the generic point of GL_{n} /H. In view of the exact sequence (26) (over F (GL_{n} /H)), the element η ∈ GL_{n} /H(F (GL_{n} /H)) gives rise to an H-torsor δ(η) = ξ ∈ H^{1}(F (GL_{n} /H), H), called the generic H-torsor. Any H-torsor is a specialization of ξ: more precisely, let F ⊂ F′ be a field extension and let x ∈ H^{1}(F′, H).

There exists a point ζ of the variety GL_{n} /H and an embedding i_{ζ} : F (ζ) → F′ such that x = i_{ζ}^{1}(δ(ζ)), where i_{ζ}^{1} : H^{1}(F (ζ), H) → H^{1}(F′, H) is the induced map in Galois cohomology.

Let H = PGO^{+}(8) ⋊ S_{3} and H ↪ GL_{n} be an embedding over F. The corresponding generic H-torsor ξ defines a trialitarian algebra T_{ξ} = (T_{ξ}, L_{ξ}, σ_{ξ}, α_{ξ}) over the function field F (GL_{n} /H).

Proposition 8.1 The algebra T_{ξ} is a division algebra.

Proof Let Z be the center of T_{ξ}. We claim that Z is a field. We may choose a specialization F_{1} of F (GL_{n} /H) with a separable cubic field extension M/F_{1}. There is a (quasi-split) trialitarian algebra T_{1} with center M (see for example [KMRT, Example (43.7)].) Specializing T to T_{1} specializes the center Z of T to M. Thus Z has to be a field. Suppose that T_{ξ} is not a division algebra, say T_{ξ} = M_{2}(A) for some central simple algebra A over Z. This implies (by specialization) that for any cubic étale algebra M_{0}/F_{0} over any field F_{0} which contains F, any trialitarian F_{0}-algebra with center M_{0} has to be of the form M_{2}(A_{0}) for some algebra A_{0} over M_{0}, which is a specialization of A. Let (D, σ) be a central division algebra of degree 8 over some field extension F_{1} of F, with an orthogonal involution σ, for example a tensor product of three generic quaternion algebras. Let Δ be the discriminant of σ (as a quadratic extension of F_{1}). In view of [KMRT, Proposition (43.15)], (D, σ) × (C(D, σ), \underline{σ}) is a trialitarian algebra over F_{1} × Δ. Since D is a division algebra, (D, σ) × (C(D, σ), \underline{σ}) is not of the form M_{2}(A_{0}). This is a contradiction. Hence T_{ξ} has to be a division algebra.

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9 Generic trialitarian division algebras

The aim of this section is to construct explicitly a generic trialitarian algebra. Our construction is inspired by and closely follows the presentation of generic division algebra given in Saltman [S2].

Let F be a field and let r be a positive integer. Consider the polynomial ring S = F [x_{ijk}, y_{ijk}, z_{ijk} | 1 ≤ i, j ≤ 8; 1 ≤ k ≤ r]. In M_{8}(S) we have the generic matrices X_{k}, Y_{k} and Z_{k} whose entries are given by (X_{k})_{ij} = x_{ijk}, (Y_{k})_{ij} = y_{ijk} and (Z_{k})_{ij} = z_{ijk}. Let U^{t} be the transpose of the matrix U and let Ũ = U − U^{t}. We recall (Section 4) that by triality, we can associate to any skew-symmetric matrix V, skew-symmetric matrices d_{ρ}(V) and d_{ρ^{2}}(V). Let D = D_{1}D_{2}D_{3} where

\begin{aligned}D_{1} & = & pf(X̃_{1} + d_{ρ}(Z̃_{1}) + d_{ρ^{2}}(Ỹ_{1})), \\ D_{2} & = & pf(Ỹ_{1} + d_{ρ}(X̃_{1}) + d_{ρ^{2}}(Z̃1)), \\ D_{3} & = & pf(Z̃_{1} + d_{ρ}(Ỹ_{1}) + d_{ρ^{2}}(X̃_{1})). \end{aligned}  (27)

Let R(F, r) be the F -subalgebra of M(S) = M_{8}(S)×M_{8}(S)×M_{8}(S) generated by all triples

(\begin{matrix}X_{k} \\ Y_{k} \\ Z_{k} \end{matrix}) , (\begin{matrix}X_{k}^{t} \\ Y_{k}^{t} \\ Z_{k}^{t} \end{matrix}) , (\begin{matrix}d_{ρ}(Z̃_{k}) + d_{ρ^{2}}(Ỹ_{k}) \\ d_{ρ}(X̃_{k}) + d_{ρ^{2}}(Z̃_{k}) \\ d_{ρ}(Ỹ_{k}) + d_{ρ^{2}}(X̃_{k}) \end{matrix}) and (\begin{matrix}(d_{ρ}(Z̃_{k}) − d_{ρ^{2}}(Ỹ_{k}))D \\ (d_{ρ}(X̃_{k}) − d_{ρ^{2}}(Z̃_{k}))D \\ (d_{ρ}(Ỹ_{k}) − d_{ρ^{2}}(X̃_{k}))D \end{matrix}) .  (28)

Let (T, L, σ_{T} , α_{T}) be a trialitarian algebra over a field extension E of F. An F -algebra homomorphism ψ: R(F, r) → T is called trialitarian if, for a standard splitting ϕ: T ⊗_{E}E′ ⟶^{∼} M(E′) of the trialitarian algebra T, there exists an F -algebra homomorphism ν: S → E′ such that M(ν): M(S) → M(E′) restricts to ψ.

Proposition 9.1 Let (T, L, σ_{T} , α_{T}) be a trialitarian algebra over a field extension E of F. Let a_{1}, a_{2}, …, a_{r} be elements of T. There exists a unique trialitarian F -algebra homomorphism ψ: R(F, r) → T such that ψ((X_{k}, Y_{k}, Z_{k})) = a_{k} for k = 1, …, r.

Proof We assume that the discriminant Δ of the cubic extension L is a field (and leave the split case as an exercise). Let E ⊂ Δ ⊂ E′ be a field extension such that (T, L, σ_{T} , α_{T})⊗_{E} E′ is isomorphic to the standard trialitarian algebra over E′, i.e., there is an isomorphism of trialitarian algebras (T, L, σ_{T} , α_{T}) ⊗_{E} E′ ⟶^{∼} M(E′). We use the isomorphism to identify T with a subalgebra of M(E′). Let

a_{k} ⊗ 1 = (A_{k}^{1}, A_{k}^{2}, A_{k}^{3}) ∈ M(E′) = M_{8}(E′) × M_{8}(E′) × M_{8}(E′)

where (A_{k}^{ϵ})_{ij} = a_{ijk}^{ϵ} ∈ E′, ϵ = 1, 2, 3. Define ϕ: S → E′ by ϕ(x_{ijk}) = a_{ijk}^{1}, ϕ(y_{ijk}) = a_{ijk}^{2} and ϕ(z_{ijk}) = a_{ijk}^{3}. Then ϕ induces a homomorphism of algebras M(ϕ) : M(S) → M(E′), where M(ϕ)((s_{ij}^{ϵ})) = (ϕ(s_{ij}^{ϵ})) for any triple of matrices (s_{ij}^{ϵ}) ∈ M_{8}(S). Let ψ be the restriction of M(ϕ) to R(F, r). We claim that ψ(R(F, r))⊂ T and that ψ is uniquely determined by the condition ψ((X_{k}, Y_{k}, Z_{k})) = a_{k} for k = 1, …, r. We check that the images under ψ of the generators of R(F, r) are in T and are determined by the a_{k}’s. Since t restricts to σ_{T} , ψ(X_{k}^{t}, Y_{k}^{t}, Z_{k}^{t}) = σ_{T} (a_{k}) ∈ T and ψ(X̃_{k}, Ỹ_{k}, Z̃_{k}) = ã_{k} = a_{k} − σ_{T} (a_{k}) ∈ T.

Let ι be the conjugation map of Δ. Since Δ ⊂ E′ we have canonical isomorphisms

Δ ⊗_{E} E′ ⟶^{∼} E′ × E′, x ⊗ y ↦ (xy, ι(x)y), x ∈ Δ and y ∈ E′, and

(T ⊗ Δ) ⊗_{E} E′ ⟶^{∼} T ⊗_{E} E′ × T ⊗_{E} E′, (a ⊗ x) ⊗ 1 ↦ (ax, aι(x))

Through this canonical isomorphism we have:

(α_{T} ∘ η_{T})(ã_{k}) ⊗_{E} 1_{E′} ↦ (\begin{matrix}d_{ρ}(Ã_{k}^{3}) & d_{ρ^{2}}(Ã_{k}^{2}) \\ d_{ρ}(Ã_{k}^{1}) & d_{ρ^{2}}(Ã_{k}^{3}) \\ d_{ρ}(Ã_{k}^{2}) & d_{ρ^{2}}(Ã_{k}^{1}) \end{matrix}) ∈ T ⊗_{E} E′ × T ⊗_{E′} E′

Further, α_{T} ∘ η_{T} (ã_{k}) ∈ ^{ρ}(Skew(T, σ) ⊗ Δ). We next compute M(ϕ)(D_{1}, D_{2}, D_{3}) and ϕ(D). Let Tr_{T ⊗_{E}Δ/T} : T ⊗_{E} Δ → T be the map induced by the trace map Δ → E and let N_{L⊗Δ/Δ} : L ⊗_{E} Δ → Δ be induced by the norm map. Since α_{T} restricts to an isomorphisms of centers Z_{T} ⟶^{∼} L ⊗_{E} Δ, L ⊗_{E} Δ ≃ L[X]/(X^{2} − d) with [d] = disc(σ_{T}) ∈ L^{×}/L^{×2}. It follows that σ_{T} ⊗_{L} 1_{L⊗_{E}Δ} is an involution of T ⊗_{L} (L ⊗_{E} Δ) = T ⊗_{E} Δ with trivial discriminant. By (6.1) there exists a reduced pfaffian Pfrd_{T ⊗_{E}Δ} : Skew(T ⊗_{E} Δ, σ_{T} ⊗ 1_{Δ}) → L ⊗_{E} Δ. We have

ψ(\begin{matrix}d_{ρ}(Z̃_{k}) + d_{ρ^{2}}(Ỹ_{k}) \\ d_{ρ}(X̃_{k}) + d_{ρ^{2}}(Z̃_{k}) \\ d_{ρ}(Ỹ_{k}) + d_{ρ^{2}}(X̃_{k}) \end{matrix}) = Tr_{T ⊗_{E}Δ/T} ((α_{T} ∘ η_{T})(ã_{k})) ∈ T.

Further M(ϕ)(D_{1}, D_{2}, D_{3}) = Pfrd_{T} (ã_{1} + Tr_{T ⊗_{E}Δ/T} ((α_{T} ∘ η_{T})(ã_{1}))) ∈ L ⊗ Δ and

ϕ(D) = N_{L⊗Δ/Δ}(Pfrd_{T} (ã_{1} + Tr_{T ⊗Δ/T} ((α_{T} ∘ η_{T})(ã_{1}))) ∈ Δ.

Let D′ = ϕ(D); since ϕ(D^{2}) = N_{L⊗Δ/Δ}(Nrd_{T} (ã_{1} + Tr_{T ⊗Δ/T} ((α_{T} ∘ η_{T})(ã_{1}))) ∈ E, it follows that ι(D′) = −D′. We have

((α_{T} ∘ η_{T})(ã_{k}) ⊗ 1_{E′})D′ = (\begin{matrix}d_{ρ}(Ã_{k}^{3})D′ & −d_{ρ^{2}}(Ã_{k}^{2})D′ \\ d_{ρ}(Ã_{k}^{1})D′ & −d_{ρ^{2}}(Ã_{k}^{3})D′ \\ d_{ρ}(Ã_{k}^{2})D′ & −d_{ρ^{2}}(Ã_{k}^{1})D′ \end{matrix}),

so that

ψ(\begin{matrix}(d_{ρ}(Z̃_{k}) − d_{ρ^{2}}(Ỹ_{k}))D \\ (d_{ρ}(X̃_{k}) − d_{ρ^{2}}(Z̃_{k}))D \\ (d_{ρ}(Ỹ_{k}) − d_{ρ^{2}}(X̃_{k}))D \end{matrix}) = T_{T ⊗_{E}Δ/T} ((α_{T} ∘ η_{T})(ã_{k}) ⊗ 1_{E′}D′) ∈ T.

Hence, as claimed, the images of the generators of R(F, r) are in T and are determined by the a_{k}’s.

□

Let E ⊂ E′ be a field extension and let

V_{T} (E′) = Hom_{F} (R(F, r), T ⊗_{E} E′)

be the set of all trialitarian F -algebra homomorphisms from R(F, r) to T ⊗_{E}E′. Let R_{L/E}(T^{r}) be the affine space T^{r} viewed as a variety over E. In view of (9.1) we have:

Corollary 9.2 V_{T} (E′) can be identified with the set of E′-points of R_{L/E}(T^{r}).

Let 0 ≠ s ∈ R(F, r). The set V_{T,s}(E′) of all trialitarian F -algebra homomorphisms ϕ : R(F, r) → T such that ϕ(s) ≠ 0 is the set of E′-points of an open subvariety V_{T,s} of V_{T}. If E′ is infinite, V_{T,s}(E′) is nonempty.

Proposition 9.3 The ring R(F, r) is a noncommutative domain.

Proof Let s, t be nonzero elements of R(F, r). Choose (T, L, σ_{T} , α_{T}) over some field E ⊃ F such that T is a division algebra. Such algebras exist in view of (8.1). Since V_{T} is irreducible, V_{T,s} ∩ V_{T,t} is an open subvariety of V_{T} and, hence, has a E-rational point ϕ: R(F, r) → T. Thus ϕ(s) and ϕ(t) are not zero and, since T is a division algebra ϕ(st) = ϕ(s)ϕ(t) ≠ 0, so that st ≠ 0.

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The center Z(F, r) of R(F, r) is a commutative domain (by (9.3)). Let L(F, r) be its field of fractions and let UT (F, r) be the ring obtained from R(F, r) by inverting all the nonzero elements of Z(F, r). Then UT (F, r) is a (noncommutative) domain with center L(F, r).

Theorem 9.4 The ring UT (F, r) is a division ring. Further it is of degree 8 over its center L(F, r).

Proof Since R(F, r) ⊂ M(S) = M_{8}(S × S × S) and since R(F, r) is a domain, R(F, r) is a prime PI-ring. By [R1, Theorem 6.1.30] UT (F, r) is central simple finite dimensional over L(F, r), hence is a central division algebra over L(F, r). To check that UT (F, r) is of degree 8 over L(F, r), we need some intermediate steps.

Proposition 9.5 1) Assume that r ≥ 3·8^{2}. There exists a basis of M(K) over K consisting of triples (X_{k}, Y_{k}, Z_{k}) ∈ R(F, r). In particular we have R(F, r)K = M(K).

2) Let Skew(K) = Skew_{8}(K) × Skew_{8}(K) × Skew_{8}(K) ⊂ M(K) be the Lie algebra of skewsymmetric elements in M(K). Assume that r ≥ 3 · 8^{2}. There is a basis of Skew(K) over K consisting of triples (X̃_{k}, Ỹ_{k}, Z̃_{k}) ∈ R(F, r).

Proof For the first claim, let {e_{ij}}, 1 ≤ i, j ≤ 8, be the standard basis of M_{8}(K), so that ((e_{ij}, 0, 0), (0, e_{ij}, 0), (0, 0, e_{ij})) is a K-basis of M(K). We have in M(K):

(X_{k}, Y_{k}, Z_{k}) = ∑_{i,j} x_{ijk}(e_{ij}, 0, 0) + ∑_{i,j} y_{ijk}(0, e_{ij}, 0) + ∑_{i,j} z_{ijk}(0, 0, e_{ij})

for k = 1, …, r. The matrix of the coefficients has generic entries, hence, if r ≥ 3 · 8^{2}, the determinant of the coefficient matrix of the first 3 · 8^{2} equations is not zero in K so that the elements {(e_{ij}, 0, 0), (0, e_{ij}, 0), (0, 0, e_{ij})} can be expressed as K-linear combinations of the triples (X_{k}, Y_{k}, Z_{k}), k ≤ 3 · 8^{2}. A similar argument gives the second claim.

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Remark 9.6 The condition r ≥ 3 · 8^{2} in (9.5) is not the best possible. Since X and X^{t} generate M_{8}(K) as a K-algebra, three triples (X_{k}, Y_{k}, Z_{k}) and their transposes generate M(K) as a K-algebra. Hence r ≥ 3 would suffice.

Corollary 9.7 1) Z(F, r) ⊂ S × S × S.

2) If (T, L, σ_{T} , α_{T}) is trialitarian and ψ: R(F, r) → T is an F -algebra homomorphism, then ψ(Z(F, r)) ⊂ L.

Proof By (9.5) R(F, r) generates M(K) over K, hence R(F, r)K = M(K). Since Z(F, r) is central in M(K), Z(F, r) is contained in K × K × K. The first claim then follows from M(S)∩(K ×K ×K) = S ×S ×S. Let ϕ : R(F, r) → T be an F -algebra homomorphism. Let T ⊗_{E} E′ ⟶^{∼} M(E′) of (T, L, σ_{T} , α_{T}) be a standard splitting. We identify (T, L, σ_{T} , α_{T}) ⊗_{E} E′ with its image in M(E′). Let ψ: R(F, r) → T ↪ M(E′) be written as (ψ^{1}, ψ^{2}, ψ^{3}) with ψ^{i}: R(F, r) → M_{8}(E′), 1 ≤ i ≤ 3, the components of ψ. We define ϕ: S → E′ by setting ϕ(x_{ijk}) = ψ^{1}(X_{k}, Y_{k}, Z_{k})_{ij}, ϕ(y_{ijk}) = ψ^{2}(X_{k}, Y_{k}, Z_{k})_{ij} and ϕ(z_{ijk}) = ψ^{3}(X_{k}, Y_{k}, Z_{k})_{ij}. The maps ψ and M(ϕ)|_{R(F,r)} give rise F -algebra homomorphisms R(F, r) → M(E′) which coincide on (X_{k}, Y_{k}, Z_{k}),1 ≤ k ≤ r. By the uniqueness statement in (9.1), ψ is the restriction of M(ϕ) to R(F, r). Since M(ϕ)(S × S × S) ⊂ L ⊗_{E} E′, we have ψ(Z(F, r)) ⊂ L ⊗_{E} E′. The second claim then follows from A ∩ (L ⊗_{E} E′) = L.

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We now prove that UT (F, r) is of degree 8 over L(F, r). Let S be the set of nonzero elements of Z(F, r). Since Z(F, r) ⊂ S × S × S, L(F, r) is contained in K × K × K. This inclusion and the inclusion R(F, r) ⊂ M(S) induce inclusion UT (F, r) = R(F, r)_{S} → M(K). We use it to identify the algebra UT (F, r) with a subalgebra of M(K). The induced map UT (F, r) ⊗_{L(F,r)} (K × K × K) → M(K) is surjective by (9.5). On the other hand UT (F, r) ⊗_{L(F,r)} (K × K × K) is central separable over K × K × K, so UT (F, r) ⊗_{L(F,r)} (K × K × K) ⟶^{∼} M(K) and UT (F, r) is of degree 8 over L(F, r).

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10 Invariants of PGO_{8}^{+}(F) ⋊ S_{3}

Let G the semidirect product PGO_{8}^{+} ⋊S_{3} as an algebraic group over F. We keep the same notation as in the previous section. In particular we set S = F [x_{ijk}, y_{ijk}, z_{ijk}]. We describe an action of G(F) on S and on M(S) = M_{8}(S) × M_{8}(S) × M_{8}(S): for b = [B] ∈ PGO_{8}(F), and U ∈ M_{8}(S) let b(U) = BUB^{−1}. Let b_{1} = \overline{ρ}(b), b_{2} = \overline{ρ}^{2}(b) ∈ PGO_{8}^{+}(F), \overline{ρ} as in (10.3). We define ϕ_{b}: S → S by

ϕ_{b}(x_{ijk}) = (b^{−1}(X_{k}))_{ij}, ϕ_{b}(y_{ijk}) = (b_{1}^{−1}(Y_{k}))_{ij} and ϕ_{b}(z_{ijk}) = (b_{2}^{−1}(Z_{k}))_{ij}.  (29)

We have, by definition of ϕ_{b}, M_{8}(ϕ_{b})(X_{k}) = b^{−1}(X_{k}), M_{8}(ϕ_{b})(Y_{k}) = b_{1}^{−1}(Y_{k}), M_{8}(ϕ_{b})(Z_{k}) = b_{2}^{−1}(Z_{k}). For ρ = (1 2 3) and π = (2 3) ∈ S_{3} we set

\begin{aligned}ϕ_{ρ}(x_{ijk}) & = & z_{ijk} \\ ϕ_{ρ}(y_{ijk}) & = & x_{ijk} \\ ϕ_{ρ}(z_{ijk}) & = & y_{ijk} \end{aligned} and \begin{aligned}ϕ_{π}(x_{ijk}) & = & (P X_{k}P)_{ij} \\ ϕ_{π}(y_{ijk}) & = & (P Z_{k}P)_{ij} \\ ϕ_{π}(z_{ijk}) & = & (P Y_{k}P)_{ij} \end{aligned}  (30)

where P ∈ GO_{8}(F) is the matrix of the conjugation π of the standard octonions with respect to the basis chosen in Section 4.

Lemma 10.1 The actions (29) and (30) combine to define an action g ↦ ϕ_{g} of G(F) on the ring S.

Proof A straightforward computation!

□

The action of G(F) on M_{8}(S) × M_{8}(S) × M_{8}(S) is as follows: let (U, V, W) ∈ M_{8}(S) × M_{8}(S) × M_{8}(S). For b ∈ PGO_{8}^{+}(F) we set

ψ_{b}(U, V, W) = b ∘ M(ϕ_{b}) = (b ∘ M_{8}(ϕ_{b})U, b_{1} ∘ M_{8}(ϕ_{b})V, b_{2} ∘ M_{8}(ϕ_{b})W ),  (31)

where b, b_{1} and b_{2} are as in (29). It is easy to check that ψ_{b}(X_{k}, Y_{k}, Z_{k}) = (X_{k}, Y_{k}, Z_{k}). For ρ and π ∈ S_{3} as above we set

ψ_{ρ}(U, V, W) = ρ ∘ M(ϕ_{ρ}) = (M_{8}(ϕ_{ρ})(V), M_{8}(ϕ_{ρ})(W), M_{8}(ϕ_{ρ})(U))  (32)

and

ψ_{π}(U, V, W) = π ∘ M(ϕ_{π}) = (P M_{8}(ϕ_{π})(U)P, P M_{8}(ϕ_{π})(W)P, P M_{8}(ϕ_{π})(V)P )  (33)

Again, it is straightforward to check that:

Lemma 10.2 The actions (31), (32) and (33) combine to give an action g ↦ ψ_{g} of G(F) on M_{8}(S) × M_{8}(S) × M_{8}(S).

We note that the above action of G(F) on M(S) restricts to an action of G(F) on the center S × S × S. In the next proposition we describe the action of G(F) on the generators of the F -algebra R(F, r):

Proposition 10.3 1) The triples (X_{k}, Y_{k}, Z_{k}) and (X_{k}^{t}, Y_{k}^{t}, Z_{k}^{t}) are invariant under the action of G(F) on M(S) = M_{8}(S) × M_{8}(S) × M_{8}(S).

2) For any U ∈ M_{8}(S), let Ũ = U − U^{t}. Let A_{3} be the alternating group with generator ρ and let G_{0}(F) = PGO_{8}^{+}(F) ⋊ A_{3}. The triple (d_{ρ}(Z̃_{k}), d_{ρ}(X̃_{k}), d_{ρ}(Ỹ_{k})) and the triple (d_{ρ^{2}}(Ỹ_{k}), d_{ρ^{2}}(Z̃_{k}), d_{ρ^{2}}(X̃_{k})) are invariant under the group G_{0}(F) and (d_{ρ}(Z̃_{k}), d_{ρ}(X̃_{k}), d_{ρ}(Ỹ_{k})) is mapped to (d_{ρ^{2}}(Ỹ_{k}), d_{ρ^{2}}(Z̃_{k}), d_{ρ^{2}}(X̃_{k})) under ψ_{π}.

3) The triple (d_{ρ}(Z̃_{k})+d_{ρ^{2}}(Ỹ_{k}), d_{ρ}(X̃_{k})+d_{ρ^{2}}(Z̃_{k}), d_{ρ}(Ỹ_{k})+d_{ρ^{2}}(X̃_{k})) is invariant under G(F).

4) The triple (D_{1}, D_{2}, D_{3}) ∈ S × S × S with

\begin{aligned}D_{1} & = & pf(X̃_{1} + d_{ρ}(Z̃_{1}) + d_{ρ^{2}}(Ỹ_{1})), \\ D_{2} & = & pf(Ỹ_{1} + d_{ρ}(X̃_{1}) + d_{ρ^{2}}(Z̃_{1})), \\ D_{3} & = & pf(Z̃_{1} + d_{ρ}(Ỹ_{1}) + d_{ρ^{2}}(X̃_{1})) \end{aligned}

is invariant under G_{0}(F) and changes sign under ψ_{π}.

5) The element D = D_{1}D_{2}D_{3} ∈ S is invariant under G_{0}(F) and changes sign under ϕ_{π}.

6) The triple ((d_{ρ}(Z̃_{k}) − d_{ρ^{2}}(Ỹ_{k}))D, (d_{ρ}(X̃_{k}) − d_{ρ^{2}}(Z̃_{k}))D, (d_{ρ}(Ỹ_{k}) − d_{ρ^{2}}(X̃_{k}))D) is invariant under G(F).

Proof Claims 1), 2) and 3) are straightforward to verify; we check that D_{1} is invariant under PGO_{8}^{+}(F): let b = [B] ∈ PGO_{8}^{+}(F); we have BB^{t} = m(B) and det(B) = m(B)^{4}. In view of (5.3) we have identities bd_{ρ} = d_{ρ}\overline{ρ}^{2}(b) and bd_{ρ^{2}} = d_{ρ^{2}} \overline{ρ}(b) and the following “small miracle” occurs:

ϕ_{b}(D_{1}) = pf(M_{8}(ϕ_{b})(X̃_{1}) + d_{ρ}(M_{8}(ϕ_{\overline{ρ}^{2}(b)})(Z̃_{1})) + d_{ρ^{2}}(M_{8}(ϕ_{\overline{ρ}(b)})(Ỹ_{1})))
= pf(b^{−1}X̃_{1} + d_{ρ}(\overline{ρ}^{2}(b)^{−1}Z̃_{1}) + d_{ρ^{2}}(\overline{ρ}(b)^{−1}Ỹ_{1}))
= pf(b^{−1}X̃_{1} + b^{−1}d_{ρ}Z̃_{1} + b^{−1}d_{ρ^{2}} Ỹ_{1})
= pf(B^{−1}(X̃_{1} + d_{ρ^{2}}(Ỹ_{1}) + d_{ρ}(Z̃_{1}))B)
= pf(m(B)^{−1}B^{t}(X̃_{1} + d_{ρ^{2}}(Ỹ_{1}) + d_{ρ}(Z̃_{1}))B)
= det(B)m(B)^{−4}D_{1} = D_{1}

The other verifications are similar. Finally 5) follows from 4) and 6) from 2), 3) and 4). □

By (10.3) the algebra R(F, r) is fixed elementwise under the action of G(F). Thus Z(F, r), L(F, r) and UT (F, r) are all fixed elementwise under this action. We shall show in (10.7) that M(K)^{G(F)} = UT (F, r) and (K × K × K)^{G(F)} = L(F, r). We start with:

Lemma 10.4 If a ∈ K^{G(F)}, then a = f/g with f, g ∈ S^{G(F)}. If u ∈ M(K)^{G(F)}, then u = v/g with v ∈ M(S)^{G(F)} and g ∈ S^{G(F)}.

Proof Since M(K) has a basis consisting of elements of R(F, r), it has a G(F)-invariant basis. Thus the second statement follows from the first. So let a ∈ K^{G(F)} and write a = f_{1}/f_{2} with f_{1}, f_{2} ∈ S without a common prime divisor. For any g = (b, β) ∈ G(F), ϕ_{g}(a) = a, so that ϕ_{g}(f_{1})f_{2} = f_{1}ϕ_{g}(f_{2}). As f_{1} and f_{2} do not have common divisors, f_{2} divides ϕ_{g}(f_{2}); ϕ_{g} being linear, we get ϕ_{g}(f_{2}) = λ(g)f_{2} for some λ(g) ∈ F^{×}. Since

ϕ_{gh}(f_{2}) = ϕ_{g}(λ(h)f_{2}) = λ(g)λ(h)f_{2}

λ defines a group homomorphism G(F) → F^{×}. For g ∈ G(F) we have g^{6} ∈ PGO_{8}^{+}(F) ⊂ PGL_{8}(F), so that λ(g)^{n} = 1 for n = 48 ([S2, Lemma 14.12]). We have a = f_{1}/f_{2} = f_{1}f_{2}^{n−1}/f_{2}^{n} with f_{2}^{n}, hence f_{1}f_{2}^{n−1} fixed under G(F).

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Our next step is to show that, if a ∈ K is invariant under G(F), then it is also invariant under G(F′) for any field extension F ⊂ F′. This is achieved by showing that a is invariant under a “generic” element of G(F). Let F [W_{ij}, N, N′] be the polynomial ring in 8^{2} + 2 variables W_{ij}, N and N′. Let W be the generic matrix with entries W_{ij} and let I be the ideal of F [W_{ij}, N, N′] defined by the relations W W^{t} = N, det(W) = N^{4} and NN′ = 1. The ring Λ = F [W_{ij}, N, N′]/I is the coordinate ring of the group PGO_{8}^{+} over F. Let w_{ij} = W_{ij} + I and (abusing notations) let W ∈ M_{8}(Λ) be the matrix (W)_{ij} = w_{ij}. The element g = ([W], γ) ∈ G(Λ) with γ arbitrary in S_{3} is the required “generic” element. We also need the notion of standard trialitarian algebra over a commutative ring. The definition over a field extends to a definition over any commutative ring R in which 2 is invertible: M(R) is the separable algebra M_{8}(R × R × R); since standard octonions are defined over R, α̃ is defined over R. We further set

PGO_{n}(R) = Aut_{R}(M_{n}(R), t) and PGO_{2l}^{+}(R) = Aut_{R}(M_{2l}(R), t, pf).  (34)

Any b ∈ PGO_{2l}^{+}(R) induces an automorphism C(b) of the Clifford algebra C(M_{2l}(R), t); however, since b, viewed as an automorphism of M_{2l}(R), needs not to be inner, we cannot use (2.1) to define C(b). The automorphism b is inner over some faithfully flat extension R′ of R (see for example [KO]). Then C(b ⊗ 1_{R′}) restricts to an automorphism of C(M_{2l}(R), t) by uniqueness. The trialitarian action of S_{3} on PGO_{8}^{+}(R) can be defined as in (5.2). In particular we set

α_{0}C(b)α_{0}^{−1} = (\overline{ρ}(b), \overline{ρ}^{2}(b)).  (35)

The definition of an automorphism of the standard trialitarian algebra over F extends to arbitrary commutative rings R as well, and we have Aut_{R}(M(R)) = PGO_{8}^{+}(R) ⋊ S_{3}. For x ∈ M(R) and g ∈ Aut_{R}(M(R)) we write g(x) = gx. For any extension ring η: S ⊂ S′ and b ∈ PGO_{8}^{+}(S′) (as defined in (34)), we define ϕ_{b}^{η}: S → S′ and ψ_{b}^{η}: M(S) → M(S′) in a similar way as in (29) and (31):

\begin{aligned}ϕ_{b}^{η}(x_{ijk}) & = & (b^{−1}(M_{8}(η)(X_{k})))_{ij} \\ ϕ_{b}^{η}(y_{ijk}) & = & (b_{1}^{−1}(M_{8}(η)(Y_{k})))_{ij} \\ ϕ_{b}^{η}(z_{ijk}) & = & (b_{2}^{−1}(M_{8}(η)(Z_{k})))_{ij} \end{aligned}  (36)

and

ψ_{b}^{η}(U, V, W) = (b(M_{8}(ϕ_{b}^{η})U), b_{1}(M_{8}(ϕ_{b}^{η})V ), b_{2}(M_{8}(ϕ_{b}^{η})W )),  (37)

where b_{1} = \overline{ρ}(b), b_{2} = \overline{ρ}^{2}(b) in PGO_{8}^{+}(S′) and \overline{ρ} is as in (35). We similarly define ϕ_{β}^{η} and ψ_{β}^{η} for any β ∈ S_{3} (see (32) and (33) for ρ and π). Let G(S′) = PGO_{8}^{+}(S′) ⋊ S_{3}. As in (10.1) and (10.2), (36) and (37) extend to maps

ϕ_{g}^{η}: S → S′ and ψ_{g}^{η} = g ∘ M(ϕ_{g}^{η}): M(S) → M(S′)  (38)

for arbitrary g ∈ G(S′).

Proposition 10.5 Let x ∈ M(S)^{G(F)} and g = ([W], γ) ∈ G(Λ) generic as above. Then, under the map ψ_{g}^{η}: M(S) → M(S ⊗ Λ) induced by the canonical embedding η: S → S ⊗ Λ, ψ_{g}^{η}(x) = M(η)(x). If θ: S → S′ is any homomorphism of commutative rings and g′ ∈ G(S′), then ψ_{g′}^{θ}(x) = M(θ)(x).

Proof The proof follows very closely the proof of [S2, Proposition 14.14]. We include it for the sake of completeness. Suppose ψ_{g}^{η}(x) ≠ M(η)(x). Let s ∈ S ⊗ Λ be a nonzero entry in the matrix triple ψ_{g}^{η}(x) − M(η)(x) ∈ M(S ⊗ Λ). Since s is a nonzero element in the polynomial ring S ⊗ Λ = Λ[x_{ijk}, y_{ijk}, z_{ijk}], there is a nonzero coefficient λ ∈ Λ of the polynomial s ∈ S ⊗ Λ. Since F is infinite, there is an F -algebra homomorphism ζ: Λ → F such that ζ(λ) ≠ 0. Let η′: : S ⊗ Λ → S be 1 ⊗ ζ. Then η′(s) is not zero since it has a nonzero coefficient ζ(λ). Let a = ζ(g) ∈ G(F). We have:

M(η′) ∘ M(ϕ_{g}^{η})(X_{k}, Y_{k}, Z_{k}) = M(ϕ_{ζ(g)})(X_{k}, Y_{k}, Z_{k})

which shows that η′ ∘ ϕ_{g}^{η} = ϕ_{ζ(g)} = ϕ_{a}. We have M(η′)(M(η)(x)) = x since η′ ∘ η = 1_{S} and

\begin{aligned}M(η′)(ψ_{g}^{η}(x) − M(η)(x)) & = & η′(g)M(η′ ∘ ϕ_{g}^{η})(x) − x \\ & = & aM(ϕ_{a})(x) − x = ψ_{a}(x) − x ≠ 0. \end{aligned}

However, since a ∈ G(F) and x ∈ M(S)^{G(F)}, we have ψ_{a}(x) = x, leading to a contradiction.

Suppose that θ: S → S′ is any ring homomorphism and let g′ ∈ G(S′). We need to show that ψ_{g′}^{θ}(x) = M(θ)(x). Since S′ is contained in a product of local rings, we may assume without loss of generality that S′ is local. If S′ is local g′ ∈ PGO_{8}^{+}(S′) is the class of an element GO_{8}^{+}(S′) modulo S′^{×} and any element of GO_{8}^{+}(S′) is the image of [W] ∈ GO_{8}^{+}(Λ) under a suitable specialization Λ → S′. Hence there is a specialization ζ: Λ → S′ such that ζ(g) = g′, where g = ([W], γ), for some γ ∈ S_{3}. Since ψ_{g}^{η}(x) = M(η)(x), where η: S → S ⊗Λ is as above and since ζθη = θ, we have ψ_{g′}^{θ}(x) = M(ζ)(M(θ)ψ_{g}^{η}(x)) = M(ζθη)(x) = M(θ)(x), as claimed.

□

The notion of trialitarian algebra extends to an arbitrary commutative ring R (where 2 is invertible). First we have an obvious definition of a pre-trialitarian algebra (T, L, σ_{T} , α_{T}) over R: L is étale cubic over R, T central separable over L, the Clifford algebra of (T, σ_{T}) is central separable over its center Z(T, σ), which is quadratic étale over R. Isomorphisms of pre-trialitarian algebras are defined as in the field case (see (25)). Let θ: R → R′ be a homomorphism of commutative rings and let T be a pre-trialitarian algebra over R. Then T ⊗_{θ} R′ is pre-trialitarian over R′. We say that T is a trialitarian algebra over R if there exists a faithfully flat morphism θ: R → S and an S-isomorphism of T ⊗_{θ} S with the standard trialitarian algebra M(S). Let (T, L, σ_{T} , α_{T}) resp. (T′, L′, σ_{T′}, α_{T′}) be trialitarian over R, resp. R′. A morphism Θ: (T, L, σ_{T} , α_{T}) → (T′, L′, σ_{T′}, α_{T′}) of trialitarian algebras is a homomorphism Θ: T → T′ of rings which restricts to a homomorphism θ: R → R′ such that Θ induces an isomorphism (T, L, σ_{T} , α_{T}) ⊗_{θ} R′ ⟶^{∼} (T′, L′, σ_{T′}, α_{T′}) of trialitarian algebras.

Proposition 10.6 Let E be a commutative F -algebra. Suppose θ_{i}: M(S) → M(E), i = 1, 2, are morphisms of trialitarian algebras such that θ_{1} and θ_{2} are equal on R(F, r). Then θ_{1} and θ_{2} are equal on M(S)^{G(F)}.

Proof Since any commutative ring embeds in a product of local commutative rings we may assume that E is local. Let e_{ij}^{ϵ}, ϵ = 1, 2, 3, and let 1 ≤ i, j ≤ 8, be the standard matrix units of M(S) = M_{8}(S) × M_{8}(S) × M_{8}(S); let f_{ij}^{ϵ} be the standard units in M(E). By definition of a morphism, θ_{ℓ}(e_{ij}^{ϵ}) = f_{ℓij}^{ϵ}, ℓ = 1, 2, form a complete set of matrix units of M(E). Let g_{1} : M(E) → M(E) be defined as the composite

M(E) ⟶^{i} M(S) ⊗_{θ_{1}} E \underrightarrow{θ̃_{1}} M(E)

where i maps the standard units f_{ij}^{ϵ} to e_{ij}^{ϵ} ⊗1 and θ̃_{1} is induced by θ_{1}. Being the composition of two morphisms, g_{1} is an automorphism of the trialitarian algebra M(E) with the property that g_{1}(f_{ij}^{ϵ}) = f_{1ij}^{ϵ}. The elements θ′_{1} = g_{1}^{−1}θ_{1} and θ′_{2} = g_{1}^{−1}θ_{2} satisfy the conditions of the proposition. If the claim is true for (θ′_{1}, θ′_{2}), the claim is also true for (θ_{1}, θ_{2}), since g_{1} is the identity on M(S)^{G(F)}. Thus, replacing θ_{1} by g_{1}^{−1}θ_{1}, we may assume that θ_{1}(e_{ij}^{ϵ}) = f_{ij}^{ϵ}, so that θ_{1} = M(η_{1}) for some ring homomorphism η_{1} : S → E. Similarly we have

θ_{2}(e_{ij}^{ϵ}) = f_{2ij}^{ϵ} = g^{−1}(f_{ij}^{ϵ})

for some g ∈ G(E) and gθ_{2} = M(η_{2}) for η_{2} : S → E. Let S′ = S ⊗_{F} E and let η′_{ℓ}: S′ → E, ℓ = 1, 2 be defined as η′_{ℓ}(s⊗t) = η_{ℓ}(s)t, i.e., η′_{ℓ} is the identity on E. We note that M(η′_{ℓ})(α) = α for any α ∈ M(E). We define θ′_{ℓ}: M(S′) = M(S) ⊗ E → M(E) as θ′_{ℓ}(α ⊗ e) = θ_{ℓ}(α)e for e ∈ E. We claim that gθ′_{2} = M(η′_{2}) on M(S′). This is clear on M(S), since gθ_{2} = M(η_{2}) on M(S), and is obvious on E, since all the maps are E-linear. Since (X_{k}, Y_{k}, Z_{k}) ∈ R(F, r) and M(η_{1}) = θ_{1}, we have

\begin{aligned}M(η′_{1})((X_{k}, Y_{k}, Z_{k})) & = & θ′_{1}((X_{k}, Y_{k}, Z_{k})) = θ′_{2}((X_{k}, Y_{k}, Z_{k})) \\ & = & g^{−1}M(η′_{2})((X_{k}, Y_{k}, Z_{k})). \end{aligned}

We have a commutative diagram

S ⊗ M(E) = M(S ⊗ E) ⟶^{M(η′_{2})=η′_{2}⊗1_{M(E)}} M(E) = E ⊗_{E} M(E)
1_{S}⊗g↓  ↓^{1_{E}⊗g}
S ⊗ M(E) = M(S ⊗ E) ⟶^{M(η′_{2})=η′_{2}⊗1_{M(E)}} M(E) = E ⊗_{E} M(E)

It follows that that g^{−1} and M(η′_{2}) commute. Now, since ψ_{g}((X_{k}, Y_{k}, Z_{k})) = (X_{k}, Y_{k}, Z_{k}), we have g^{−1}((X_{k}, Y_{k}, Z_{k})) = M(ϕ_{g})((X_{k}, Y_{k}, Z_{k})). It follows that M(η′_{1})((X_{k}, Y_{k}, Z_{k})) = M(η′_{2}ϕ_{g})((X_{k}, Y_{k}, Z_{k})) hence η′_{1} = η′_{2}ϕ_{g}. We then have for any β ∈ M(S′):

\begin{aligned}θ′_{1}(β) & = & M(η′_{1})(β) = M(η′_{2})M(ϕ_{g})(β) \\ & = & M(η′_{2})g^{−1}gM(ϕ_{g})(β) = g^{−1}M(η′_{2})ψ_{g}(β) = θ′_{2}(ψ_{g}(β)), \end{aligned}

so that θ′_{1} = θ′_{2}ψ_{g}. Choose α ∈ M(S)^{G(F)}. By (10.5) ψ_{g}(α) = α and θ_{1}(α) = θ′_{1}(α) = θ′_{2}(α) = θ_{2}(α), hence the claim.

□

We are now ready for the description of UT (F, r) and L(F, r) as invariant rings:

Theorem 10.7 Let G(F) = PGO_{8}^{+}(F)⋊S_{3}. We have UT (F, r) = M(K)^{G(F)} and L(F, r) = (K × K × K)^{G(F)}.

Proof It suffices to prove the first claim, since the second follows by restricting the action to the center. We know already that UT (F, r) ⊂ M(K)^{G(F)}. Set UT = UT (F, r), L = L(F, r) and K^{3} = K × K × K. Define η_{ℓ}: K^{3} → K^{3} ⊗_{L} K^{3}, ℓ = 1, 2, by η_{1}(x) = x ⊗ 1 and η_{2}(x) = 1⊗x. Since L is a field, it is clear that for x ∈ K^{3}, x ∈ L if and only if η_{1}(x) = η_{2}(x). We set θ_{ℓ} = M_{8}(η_{ℓ}): M_{8}(K^{3}) → M_{8}(K^{3} ⊗_{L} K^{3}), ℓ = 1, 2. We have M_{8}(K^{3}) = M(K) and M_{8}(K^{3} ⊗_{L} K^{3}) = M(K)⊗_{L} K^{3}. We identify M(K) with UT (F, r)⊗_{L} K^{3} and M_{8}(K^{3} ⊗_{L} K^{3}) with UT (F, r) ⊗_{L} (K^{3} ⊗_{L} K^{3}). Then the θ_{ℓ} are maps

θ_{l} : UT (F, r) ⊗_{L} K^{3} → UT (F, r) ⊗_{L} (K^{3} ⊗_{L} K^{3})

and, for α ∈ M(K), α ∈ UT (F, r) if and only if θ_{1}(α) = θ_{2}(α). Let α ∈ M(K)^{G(F)} and write (by (10.4)) α = β/h with β ∈ M(S)^{G(F)} and h ∈ S^{G(F)}. Thus it suffices to show that M(S)^{G(F)} ⊂ UT (F, r). Since the θ_{ℓ} are equal on UT (F, r) they are equal on R(F, r). By (10.6) they are equal on M(S)^{G(F)}, hence M(S)^{G(F)} ⊂ UT (F, r).

□

We still need to define a trialitarian structure on the central division algebra UT (F, r) over L(F, r), in particular we need a base field k(F, r) over which L(F, r) is cubic separable field extension. In view of (10.7) an obvious candidate is the field k(F, r) = K^{G(F)}. We start with an intermediate result:

Lemma 10.8 1) Let D ∈ S be as in (10.3), 5); then D ∉ k(F, r) and D^{2} ∈ k(F, r), so that k(F, r)[D] ⊂ K is a quadratic separable extension of k(F, r).

2) Let L(F, r)[D] = L(F, r) ⊗_{k(F,r)} k(F, r)[D] ⊂ K^{3}. Let ρ̃: K^{3} → K^{3} be the cyclic permutation (x, y, z) ↦ (z, x, y). Then ρ̃(L(F, r)[D]) ⊂ L(F, r)[D] and L(F, r)[D]/k(F, r)[D] is a cubic extension, which is cyclic, with the restriction of ρ̃ to L(F, r)[D] as a generator of the Galois group.

Proof The first claim follows from (10.3), 5). We prove 2): Let ρ = (1 2 3) ∈ S_{3}. Since ψ_{ρ} leaves D fixed (see (10.3), 5)), L(F, r)[D] is invariant under G_{0}(F) = PGO_{8}^{+}(F) ⋊ A_{3}. Since D ∉ L(F, r), [L(F, r)[D] : L(F, r)] = 2. It then follows from (G(F) : G_{0}(F)) = 2 that L(F, r)[D] = (K × K × K)^{G_{0}(F)}. Let (x, y, z) ∈ L(F, r)[D] ⊂ K × K × K. We show that ρ̃((x, y, z)) is invariant under G_{0}(F). Let g = g_{1}ρ′, g_{1} ∈ PGO_{8}^{+}(F) and ρ′ = (1 2 3) ∈ A_{3}. Since L(F, r)[D] is invariant under G_{0}(F), we have

\begin{aligned}(x, y, z) & = & ψ_{g_{1}ρ′}(x, y, z) = ψ_{g_{1}}(ρ′(ϕ_{ρ′}x, ϕ_{ρ′}y, ϕ_{ρ′}z)) = ψ_{g_{1}}(ϕ_{ρ′}z, ϕ_{ρ′}x, ϕ_{ρ′}y) \\ & = & (g_{1}ϕ_{ρ′}(z), g_{1}ϕ_{ρ′}(x), g_{1}ϕ_{ρ′}(y)) \end{aligned}

so x = g_{1}ϕ_{ρ′}(z), y = g_{1}ϕ_{ρ′}(x) and z = g_{1}ϕ_{ρ′}(y). On the other hand we have

ψ_{g_{1}ρ′}(ρ̃(x, y, z)) = (g_{1}ψ_{ρ′}(y), g_{1}ψ_{ρ′}(z), g_{1}ψ_{ρ′}(x)) = (z, x, y) = ρ̃(x, y, z).

The same computation holds for g_{1}ρ′^{−1}, so that ρ̃(x, y, z) ∈ L(F, r)[D] and ρ̃(L(F, r)[D]) ⊂ (L(F, r)[D]). Replacing ρ̃ by ρ̃^{−1} we see that ρ̃ is an automorphism of L(F, r)[D]. We claim that k(F, r)[D] = L(F, r)[D]ρ̃ is the fixed field of L(F, r)[D] under ρ̃. The field k(F, r)[D] is clearly invariant under ρ̃. For the other inclusion, we recall first that L(F, r)[D] = (K × K × K)^{G_{0}(F)}. We show that k(F, r)[D] = K^{G_{0}(F)}: clearly k(F, r)[D] ⊂ K^{G_{0}(F)} and (by definition) k(F, r) = K^{G(F)}. Since D ∉ k(F, r) and D^{2} ∈ k(F, r) we have [k(F, r)[D] : k(F, r)] = 2, so k(F, r)[D] = K^{G_{0}(F)}. Now, if x ∈ L(F, r)[D] is invariant under ρ̃, then x ∈ L(F, r)[D] ∩ K ⊂ K^{G_{0}(F)} = k(F, r)[D].

□

Theorem 10.9 The extension L(F, r)/k(F, r) is of degree 3 and has discriminant D^{2}.

Proof The first claim follows from [k(F, r)[D] : k(F, r)] = [L(F, r)[D] : L(F, r)] = 2 and [L(F, r)[D] : k(F, r)[D]] = 3. The last claim holds since by adjoining D, L(F, r)[D] over k(F, r)[D] becomes cyclic.

□

Theorem 10.10 Let Δ(F, r) = k(F, r)[D]. There is an orthogonal involution σ_{UT} on UT (F, r) and an isomorphism

α_{UT} : C(UT (F, r), σ_{UT} ) ⟶^{∼} ρ̃(UT (F, r) ⊗_{k(F,r)} Δ(F, r))

such that (UT (F, r), L(F, r), σ_{UT} , α_{UT} ) is a trialitarian algebra over k(F, r).

Proof Let M(K) be the standard trialitarian structure over K, with isomorphism

α̃: C(M(K), t) ⟶^{∼} ^{ρ}M(K) × ^{ρ^{2}}M(K).

Let η: Skew(M(K), t) → C(M(K), t) be the canonical embedding (15). The isomorphism α̃ is uniquely determined by the condition

α̃ ∘ η((a, b, c)) = ((d_{ρ}(c), dρ(a), d_{ρ}(b)), (d_{ρ^{2}}(b), d_{ρ^{2}}(c), d_{ρ^{2}}(a)))  (39)

for any (a, b, c) ∈ Skew(M(K), t). We set UT = UT (F, r). Since UT is a subalgebra of M(K) which generates M(K) over K we have UT ⊗_{k(F,r)} K ⟶^{∼} M(K). The transpose t on M(K) maps R(F, r) to itself and hence restricts to an involution σ_{UT} on UT and C(UT, σ_{UT} ) is a subalgebra of the Clifford algebra C(M(K), t); we set α_{UT} = α|_{UT}. To check that α_{UT} defines a trialitarian structure on UT, it suffices to verify that α_{UT} ∘ η(Skew(UT, σ_{UT})) ⊂ ρ̃(UT ⊗_{k(F,r)} Δ). Since Skew(UT, σ_{UT}) is (linearly) generated over L(F, r) by (X̃_{k}, Ỹ_{k}, Z̃_{k}) (see (9.5)), it suffices to check that the images of the (X̃_{k}, Ỹ_{k}, Z̃_{k}) belong to ρ̃(UT ⊗_{k(F,r)} Δ). The embedding UT ⊂ M(K) induces an embedding

ρ̃(UT ⊗_{k(F,r)} Δ(F, r)) ⊂ ^{ρ}(M(K) ⊗_{k(F,r)} Δ(F, r)).

We may choose D^{−1} as a generator of Δ(F, r), instead of D. Since D ∈ K, we have an identification M(K) ⊗_{k(F,r)} Δ(F, r) ⟶^{∼} M(K) × M(K) given by a ⊗ 1 + b ⊗ D^{−1} ↦ (a + bD^{−1}, a − bD^{−1}), so that for a_{1}, a_{2} ∈ M(K):

\frac{a_{1} + a_{2}}{2} ⊗ 1 + \frac{(a_{1} − a_{2})}{2}D ⊗ D^{−1} ↦ (a_{1}, a_{2}).  (40)

With

a_{1} = (d_{ρ}(Z̃_{k}), d_{ρ}(X̃_{k}), d_{ρ}(Ỹ_{k}))^{t} and a_{2} = (d_{ρ^{2}}(Ỹ_{k}), d_{ρ^{2}}(Z̃_{k}), d_{ρ^{2}}(X̃_{k}))^{t}

in (40), (39) implies that the triple (X̃_{k}, Ỹ_{k}, Z̃_{k}) maps to

\frac{1}{2} (\begin{matrix}d_{ρ}(Z̃_{k}) + d_{ρ^{2}}(Ỹ_{k}) \\ d_{ρ}(X̃_{k}) + d_{ρ^{2}}(Z̃_{k}) \\ d_{ρ}(Ỹ_{k}) + d_{ρ^{2}}(X̃_{k} \end{matrix}) + \frac{1}{2} (\begin{matrix}(d_{ρ}(Ỹ_{k}) − d_{ρ^{2}}(Z̃_{k}))D \\ (d_{ρ}(Z̃_{k}) − d_{ρ^{2}}(X̃_{k}))D \\ (d_{ρ}(X̃_{k}) − d_{ρ^{2}}(Ỹ_{k}))D \end{matrix})

This element is in ^{ρ}(UT ⊗_{k(F,r)} Δ), since it is the sum of elements in R(F, r), hence the claim.

□

We call UT (F, r) the generic trialitarian algebra.

Remark 10.11 The association F ↦ UT (F, r) is functorial: any injection F → F′ of fields yields in a natural way a morphism UT (F, r) → UT (F′, r) of trialitarian algebras.

Remark 10.12 The field k(F, r) has transcendental dimension 3r − 1 + 36. We plan to give some properties of k(F, r) in a forthcoming paper.

11 Specialization

Let T = (T, L, σ_{T} , α_{T}) be a trialitarian algebra over a field k and T′ = (T′, L′, σ_{T′}, α_{T′}) be a trialitarian algebra over a field k′. We say that T specializes to T′ if there is a domain R ⊂ k and a trialitarian algebra (B, L_{B}, σ_{B}, α_{B}) over R such that the following conditions are satisfied:

1) k is the field of fractions of R and there is an isomorphism of trialitarian algebras B⊗_{R}k ≃ T;

2) there is a homomorphism ϕ: R → k′ inducing an isomorphism B ⊗_{ϕ}k′ ⟶^{∼} T′ of trialitarian algebras over k′.

In the rest of the section we give a proof of the following

Theorem 11.1 For any trialitarian algebra (T, L, σ_{T} , α_{T}) over any field extension k ⊃ F, there exists a specialization of the generic trialitarian algebra (UT (F, r), L(F, r), σ_{UT} , α_{UT}) to (T, L, σ_{T} , α_{T}).

The proof of the theorem is in several steps. The first step is to lift the trialitarian algebra (UT (F, r), L(F, r), σ_{UT} , α_{UT}) to a trialitarian algebra over a commutative ring whose field of fractions is k(F, r). We set V_{s} = V ⊗_{R} R[1/s] for a module V over a domain R and a nonzero element s ∈ R.

Lemma 11.2 Let (T, L, σ, α) be a trialitarian algebra over a field k. Let B ↪ T, S ↪ L, R ↪ k, δ ↪ Δ be subrings such that B ⊃ S ⊃ R, δ ⊃ R and such that k is the quotient field of R; further assume that B is central separable over S of degree 8, S is cubic étale over R and δ is quadratic étale over R. Then there exists s ∈ R, s ≠ 0, such that σ restricts to an involution of B_{s}, ρ on L ⊗ δ restricts to an automorphism of S_{s} ⊗ δ_{s} and α restricts to an isomorphism C(B_{s}, σ) ⟶^{∼} ^{ρ}((B_{s}, σ) ⊗_{R_{s}} δ_{s}); i.e., (B_{s}, S_{s}, σ, α) is a trialitarian algebra over R_{s} with the natural map (B_{s}, S_{s}, σ, α) ⊗_{R_{s}} k ⟶^{∼} (T, L, σ, α) an isomorphism of trialitatrian algebras over k.

Proof The lemma is immediate, observing that B, S and δ are finitely generated as Rmodules (in particular Sk = L, δk = Δ and Bk = T).

□

From now on we use the same notations R(F, r), Z(F, r), UT (F, r), L(F, r) and k(F, r) as in Section 9.

Proposition 11.3 There exists an element s ∈ Z(F, r), s ≠ 0, and F -algebras E ⊂ Z(F, r)_{s}, δ ⊂ Z(F, r)_{s}[D], δ ⊃ E, such that:

1) R(F, r)_{s} is Azumaya over Z(F, r)_{s},

2) Z(F, r)_{s} is cubic étale over E,

3) δ is quadratic étale over E,

4) k(F, r) is the quotient field of E.

Proof The ring R(F, r) is a prime PI-ring (see the proof of (9.4)). By [R1, 6.3.27] there exists s ∈ Z(F, r) such that R(F, r)_{s} is central separable over Z(F, r)_{s}. Since R(F, r)_{s} is central separable over Z(F, r)_{s} and is finitely generated as an F -algebra, Z(F, r)_{s} is a finitely generated F -algebra. Let D be as in (27); since D^{2} ∈ k(F, r) ↪ L(F, r) we may assume, after inverting a further element of Z(F, r), that D^{2} ∈ Z(F, r)_{s}. Then the ring Z(F, r)_{s}[D] is a finitely generated F -algebra with quotient field L(F, r)[D]. Let ρ̃ be the automorphism of L(F, r)[D] as in (10.8) and let π̃ be the automorphism sending D to −D and fixing L(F, r). Then ρ̃ and π̃ generate a subgroup S̃_{3} of Aut_{k(F,r)}(L(F, r)[D]) isomorphic to S_{3} and with fixed field k(F, r). Because of the finite generation of Z(F, r)_{s} over F, one may invert a further element of Z(F, r)_{s} and assume that S̃_{3} restricts to an action on Z(F, r)_{s}[D]. We set E = (Z(F, r)_{s}[D])S̃_{3}, δ = (Z(F, r)_{s})ρ̃. Then E ⊂ Z(F, r)_{s}[D]π̃ = Z(F, r)_{s}. We claim that

k(F, r) is the quotient field of E. Since (Z(F, r)_{s}[D])S̃_{3} = E, the norm map L(F, r) → k(F, r) restricts to a norm map N: Z(F, r)_{s}[D] → E. Let y ∈ k(F, r), y = a/b, a, b ∈ Z(F, r)_{s}[D]; then y = a(Nb)b^{−1}(Nb)^{−1}, with Nb ∈ E. The element a(Nb)b^{−1} = a·∏_{σ∈S̃_{3}, σ≠1} σ(b) belongs to Z(F, r)_{s}[D] and, since y is invariant under S̃_{3}, yNb = aNbb^{−1} is invariant under S̃_{3}. Hence aNbb^{−1} belongs to E. Thus k(F, r) is the quotient field of E. Since Z(F, r)_{s}[D]S̃_{3} = E, Z(F, r)_{s}[D]ρ̃ = δ; δ and Z(F, r)_{s} are finitely generated F -algebras which are integral over E. Hence they are finite as E-modules and we may choose s′ ∈ E such that Z(F, r)_{ss′} and δ_{s′} are étale over E_{s′}. Replacing E_{s′} by E, the proposition follows from (11.2).

□

We thus have proved the following

Theorem 11.4 There exists a nonzero element s ∈ Z(F, r) such that the trialitarian algebra (UT (F, r), L(F, r), σ_{UT} , α_{UT} ) restricts to a trialitarian algebra (R(F, r)_{s}, Z(F, r)_{s}, \overline{σ}, \overline{α}) over E ⊂ k(F, r), the quotient field of E being k(F, r), such that the natural map R(F, r)_{s} ⊗_{E} k(F, r) → UT (F, r) is an isomorphism of trialitarian algebras over k(F, r).

We next show that for any field extension F ⊂ k and any trialitarian algebra (T, L, σ, α) over k, there is a specialization of (R(F, r)_{s}, Z(F, r)_{s}, \overline{σ}, \overline{α}) to (T, L, σ, α).

Let N: K × K × K → K be the norm, i.e., N(x, y, z) = xyz for x, y and z ∈ K. Let s ∈ Z(F, r) be as in (11.4). Since Z(F, r)_{s} ⊂ (S × S × S)_{s}, we have E ⊂ S_{Ns}. We further have s_{0} ∈ S such that R(F, r) · S_{s_{0}} = M(S_{s_{0}}) (see the proof of (9.5)); in particular R(F, r)_{s} · S_{Ns·s_{0}} = M(S_{Ns·s_{0}}) and the natural map θ: R(F, r)_{s} ⊗_{E} S_{Ns·s_{0}} → M(S_{Ns·s_{0}}) is an isomorphism of S_{Ns·s_{0}}-algebras. Since the restriction of the standard triality on M(K) to M(S_{Ns·s_{0}}) is the standard triality on this algebra and since its restriction to R(F, r)_{s} is the restriction of the triality on UT (F, r) to R(F, r)_{s}, θ is an isomorphism of trialitarian algebras. Thus we have proved:

Proposition 11.5 The natural map θ: R(F, r)_{s} ⊗_{E} S_{Ns·s_{0}} → M(S_{Ns·s_{0}}) is an isomorphism of trialitarian algebras.

Let (T, L, σ, α) be a trialitarian algebra over k ⊃ F. We recall (see (9.2)) that X = Hom_{F} (R(F, r), T ) is an affine space R_{L/F} (T^{r}). Let k ⊂ k′ be such that there is an isomorphism T ⊗_{k} k′ ≃ M(k′) of trialitarian algebras. We regard T as a subalgebra of M(k′) through this isomorphism. We view any ψ ∈ X as a map R(F, r) → T ↪ M(k′). Let ϕ_{ψ}: S → k′ be such that M(ϕ_{ψ}) restricted to R(F, r) is ψ. We recall that

ϕ_{ψ}(x_{ijk}) = a_{ijk}, ϕ_{ψ}(y_{ijk}) = b_{ijk} and ϕ_{ψ}(z_{ijk}) = c_{ijk}

if

ψ((X_{k}, Y_{k}, Z_{k})) = ((a_{ijk}), (b_{ijk}), (c_{ijk}))∈ T ↪ M(k′), 1 ≤ i, j ≤ 8.

Further, given an r-tuple \underline{a} = (a_{1}, a_{2}, …, a_{r}) of elements of T ↪ M(k′), let ψ\underline{a}: R(F, r) → T (cf (9.1))be the restriction of M(ϕ\underline{a}), where

ϕ\underline{a}(x_{ijk}) = (A_{k}^{1})_{ij}, ϕ\underline{a}(y_{ijk}) = (A_{k}^{2})_{ij}, ϕ\underline{a}(z_{ijk}) = (A_{k}^{3})_{ij},

for a_{k} = (A_{k}^{1}, A_{k}^{2}, A_{k}^{3}) ∈ M(k′), 1 ≤ k ≤ r. We already remarked that the following subvarieties of X are open:

U_{s} = {ψ ∈ X | ψ(s) ∈ T^{×}}, s ∈ Z(F, r)
V = \{ψ\underline{a} | (a_{1}, …, a_{r}), with (a_{1}, …, a_{ℓ}), ℓ = 3 · 8^{2} ≤ r, a k-basis of T \}.

Let s_{0} ∈ S be as in (11.5) and let W = {ψ ∈ X | ϕ_{ψ}(s_{0}) ≠ 0}. Clearly X = Hom_{F} (R(F, r), T ) is identified with a subvariety of Hom_{F} (S, k′) under the map ψ ↦ ϕ_{ψ} and the set {ϕ ∈ Hom_{F} (S, k′), ϕ(s_{0}) ≠ 0} is open in Hom_{F} (S, k′). Hence W is open in X. Let s be as in (11.3). The intersection of the three open subsets U_{s}, V and W of X is not empty (since F is infinite), hence contains a k-rational point ψ\underline{a}. The element \underline{a} = (a_{1}, a_{2}, …, a_{r}) is such that (a_{1}, a_{2}, …, a_{ℓ}), ℓ = 3 · 8^{2} ≤ r, is a basis of T over k, ψ\underline{a}((X_{k}, Y_{k}, Z_{k})) = a_{k}, 1 ≤ k ≤ r, ϕ_{ψ\underline{a}}(s_{0}) ≠ 0 in k′ and ψ\underline{a}(s) is a unit of T. We abbreviate ψ\underline{a} = ψ and ϕ_{ψ\underline{a}} = ϕ. The homomorphism ϕ: S → k′ has a nonzero value on N(s) · s_{0}, hence yields a homomorphism ϕ: S_{Ns·s_{0}} → k′. Since E ⊂ S_{Ns} ∩ R(F, r), ϕ restricts to ϕ: E → k, because M(ϕ)(R(F, r)) = ψ(R(F, r)) and T ∩ k′ = k. The homomorphism ψ: R(F, r) → T extends to ψ: R(F, r)_{s} → T since ψ(s) is a unit of T. Since ψ is the restriction of M(ϕ): M(S_{s}) → M(k′), ψ(Z(F, r)_{s}) ⊂ L and ψ(E) ⊂ k (see (11.3)). We claim that the map ψ̃: R(F, r)_{s} ⊗_{ϕ} k → T induced by ψ is an isomorphism of trialitarian algebras. First, ψ̃ is an isomorphism of k-algebras since the k-basis (a_{1}, …, a_{ℓ}) is caught in the image of ψ̃. To verify that it is an isomorphism of trialities, it suffices to check this after base change to k′. The composite

R(F, r)_{s} ⊗_{ϕ} k ⊗_{k} k′ ⟶^{∼} R(F, r)_{s} ⊗_{ϕ} k′ ≃

R(F, r)_{s} ⊗_{E} S_{Ns·s_{0}} ⊗_{ϕ} k′ ⟶^{∼} M(S_{Ns·s_{0}}) ⊗_{ϕ} k′ ≃ M(k′)

is precisely ψ̃ ⊗ 1. The isomorphism R(F, r)_{s} ⊗_{E} S_{Ns·s_{0}} ⟶^{∼} M(S_{Ns·s_{0}}) is an isomorphism of trialities. Hence each arrow in the above composition is an isomorphism of trialities and so is ψ̃ ⊗ 1. Thus we have proved:

Theorem 11.6 Let (T, L, σ, α) be a trialitarian algebra over k ⊃ F. With the notation as above, there is an homomorphism ϕ: E → k and an homomorphism ψ: R(F, r)_{s} → T of F -algebras such that ψ|_{E} = ϕ and ψ̃: R(F, r)_{s} ⊗_{ϕ} k → T is an isomorphism of trialitarian algebras. In particular the trialitarian algebra (UT (F, r), L(F, r), σ_{UT} , α_{UT} ) admits a specialization.

Remark 11.7 Let Δ be the discriminant of the cubic extension L(F, r) (as a quadratic algebra). Then L(F, r)⊗Δ is cubic cyclic over Δ, UT (F, r)⊗Δ = M(K)^{PGO_{8}^{+} ⋊A_{3}} is a generic cyclic trialitarian algebra and admits a specialization to any cyclic trialitarian algebra.

12 Generic Orthogonal and Unitary Involutions

Following the pattern used in constructing the generic trialitarian algebra UT (F, r), it is straightforward to construct a generic algebra with an involution of orthogonal type and trivial discriminant, as well as a generic algebra with a unitary involution (i.e., of the second kind). We first describe the orthogonal case. Let S = F [x_{ijk}], 1 ≤ i, j ≤ n, k = 1, …, r, r ≥ n^{2}, with field of fractions K. Let X_{k}, k = 1, …, r, be the generic matrix with entries (X_{k})_{ij} = x_{ijk}; Let R(F, n, r) be the F -subalgebra of M_{n}(K) generated by the generic matrices X_{k} and their transposes X_{k}^{t}. Let Z(F, n, r) be the center of R(F, n, r). The rings R(F, n, r) and Z(F, n, r) are domains. Let k(F, n, r) be the quotient field of Z(F, n, r) and U(F, n, r) be the central localization of R(F, n, r), i.e., the localization with respect to all nonzero elements of Z(F, n, r). The algebra U(F, n, r) is a central simple algebra over k(F, n, r). The transpose on M_{n}(K) restricts to an involution, denoted σ, of R(F, n, r), hence of U(F, n, r). The involution σ restricts to the identity on the center k(F, n, r) and is of orthogonal type. The pair (U(F, n, r), σ) is the generic central simple algebra with orthogonal involution considered by Rowen [R1]. On the same lines as in the proof of (10.7) we get Procesi’s result [P2]:

U(F, n, r) = M_{n}(K)^{PGO_{n}(F)} and k(F, n, r) = K^{PGO_{n}(F)}

We assume from now on that n = 2l is even. The matrix X_{1} − X_{1}^{t} is skew-symmetric; let D = pf(X_{1} − X_{1}^{t}). The element D^{2} = det(X_{1} − X_{1}^{t}) belongs to k(F, 2l, r); further D ∉ k(F, 2l, r), otherwise disc(σ) = 1, but σ specializes to involutions which do not have trivial discriminant. Thus Δ(F, 2l, r) = k(F, 2l, r)[D] ⊂ K is a quadratic field extension of k(F, 2l). Let Ũ(F, 2l, r) = U(F, 2l, r) ⊗_{k(F,2l,r)} Δ(F, 2l, r) and σ̃ = σ ⊗ 1_{Δ(F,2l,r)}; then (Ũ(F, 2l, r), σ̃) is the generic central simple algebra of degree 2l with involution of orthogonal type with trivial discriminant; this generic algebra admits a specialization to any central simple algebra of degree 2l with involution of orthogonal type with trivial discriminant. The algebra Ũ(F, 2l, r) is the invariant ring of M_{2l}(K) under the action of PGO_{2l}^{+}(F) and Δ(F, 2l, r) is the field of invariants for the same action restricted to K.

We finally consider the case of unitary involutions. Let, as above, S = F [x_{ijk}, y_{ijk}], 1 ≤ i, j ≤ n, k = 1, …, r, with field of fractions K. Let R^{u}(F, n, r) be the F -subalgebra of M_{n}(K) × M_{n}(K) generated by the pairs (X_{k}, Y_{k}), (Y_{k}^{t}, X_{k}^{t}). Let Z^{u}(F, n, r) be the center of R^{u}(F, n, r). The rings R^{u}(F, n, r) and Z^{u}(F, n, r) are domains. Let K(F, n, r) be the quotient field of Z^{u}(F, n, r) and let U^{u}(F, n, r) be the central localization of R^{u}(F, n, r) with respect to all nonzero elements of Z^{u}(F, n, r). The algebra U^{u}(F, n, r) is a central division algebra over Z(F, n, r). The involution τ: (x, y) ↦ (y^{t}, x^{t}) of M_{n}(K) × M_{n}(K) is of second kind and restricts to an involution (also denoted τ) of R^{u}(F, n, r), hence of U^{u}(F, n, r). It further restricts to an automorphism ι of the center Z^{u}(F, n, r) of order 2. Let k(F, n, r) be the elements of Z^{u}(F, n, r) fixed under ι. Then (U^{u}(F, n, r), τ) is a generic division algebra with unitary involution, in the sense that it admits a specialization to any central simple algebra with unitary involution over a field k which contains F. Let further G(F) = PGL_{n}(F) ⋊ ℤ/2ℤ, where the action of ℤ/2ℤ on PGL_{n}(F) is through x ↦ (x^{t})^{−1} for x ∈ GL_{n}(F). The group G(F) acts on S and on M_{n}(S) × M_{n}(S), hence on K and M_{n}(K) × M_{n}(K) and (M_{n}(K) × M_{n}(K))^{PGL_{n}(F)⋊ℤ/2ℤ} = U^{u}(F, n, r), (K × K)^{PGL_{n}(F)⋊ℤ/2ℤ} = Z^{u}(F, n, r) and K^{PGL_{n}(F)⋊ℤ/2ℤ} = k(F, n, r).

Remark 12.1 To construct the generic algebra with unitary involution, we may also start with a half-split datum: let F_{1} = F (\sqrt{u}) be a generic quadratic extension, i.e., u is an indeterminate, and let ι: F_{1} → F_{1} be the F -linear map with ι(\sqrt{u}) = −\sqrt{u}. Let K_{1} = K ⊗_{F} F_{1}, where K = F (x_{ijk}), and let R(F_{1}, n, r)) be the F_{1}-subalgebra of M_{n}(K_{1}) generated by generic matrices X_{k} and their transposes X_{k}^{t}. The unitary involution τ of M_{n}(K_{1}) which restricts to 1_{K} ⊗ ι on the center and maps X_{k} to X_{k}^{t} restricts to a unitary involution of R(F_{1}, n, r). Let K^{∗}(F, n.r) be the center of U^{∗}(F, n, r). The group PGU_{n}(F) = {x ∈ PGL_{n}(F_{1}) | xτ(x) ∈ F_{1}^{×}}; acts on M_{n}(K_{1}) and M_{n}(K_{1})^{PGU_{n}(F)} = U^{∗}(F, n, r), K_{1}^{PGU_{n}(F)} = K^{∗}(F, n, r).

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