CONJUGACY CLASSES OF TRIALITARIAN AUTOMORPHISMS AND SYMMETRIC COMPOSITIONS

V. CHERNOUSOV, M.-A. KNUS, AND J.-P. TIGNOL

Dedicated with great friendship to Eva Bayer on the occasion of her 60^{th} birthday

Abstract. The trialitarian automorphisms considered in this paper are the outer automorphisms of order 3 of adjoint classical groups of type D_{4} over arbitrary fields. A one-to-one correspondence is established between their conjugacy classes and similarity classes of symmetric compositions on 8-dimensional quadratic spaces. Using the known classification of symmetric compositions, we distinguish two conjugacy classes of trialitarian automorphisms over algebraically closed fields. For type I, the group of fixed points is of type G_{2}, whereas it is of type A_{2} for trialitarian automorphisms of type II.

1. Introduction

Among simple algebraic groups of classical type only the simple adjoint algebraic groups G = PGO^{+}(n) and the simple simply connected algebraic groups G = Spin(n), where n is the norm of an octonion algebra, admit outer automorphisms of order 3, known as trialitarian automorphisms, see [KMRT98, (42.7)] or [Jac64]. These groups are of type D_{4} and there is a split exact sequence of algebraic groups

(1.1)  1 → Int(G) → Aut(G) → 𝔖_{3} → 1

where the permutation group of three elements 𝔖_{3} is viewed as the group of automorphisms of the Dynkin diagram of type D_{4}:

[figure]

In view of the exact sequence (1.1), all the trialitarian automorphisms of G can be obtained from a fixed one ρ by composing ρ or ρ^{−1} with inner automorphisms. They are not necessarily conjugate to ρ, however. Our goal is to classify trialitarian automorphisms defined over an arbitrary field F up to conjugation in the group Aut(G)(F) of F -automorphisms of G. We achieve this goal by relating trialitarian automorphisms to symmetric compositions and using the known classification of composition algebras.

Date: June 25, 2011.

2010 Mathematics Subject Classification. 20G15, 11E88.

Key words and phrases. Triality, composition algebras, octonions.

The first author was partially supported by the Canada Research Chairs Program and an NSERC research grant, the third author by the F.R.S.–FNRS (Belgium).

We take as guiding principle the analogous description of outer automorphisms of order 2 of PGL_{n} for n ≥ 3, which could be termed dualitarian automorphisms: they have the form f ↦ σ(f)^{−1}, where σ is the adjoint involution of a nonsingular symmetric or skew-symmetric bilinear form. Dualitarian automorphisms on PGL_{n} are thus in one-to-one correspondence with nonsingular symmetric or skew-symmetric bilinear forms on an n-dimensional vector space up to scalar multiples, and dualitarian automorphisms are conjugate if and only if the corresponding bilinear forms are similar. There are two types of dualitarian automorphisms, distinguished by the type of their groups of fixed points, which can be either symplectic or (in characteristic different from 2) orthogonal. In characteristic 2, the non-symplectic case leads to group schemes that are not smooth.

Likewise, we set up a one-to-one correspondence between trialitarian automorphisms of PGO^{+}(n), for n a 3-fold Pfister quadratic form, and symmetric compositions up to scalar multiples on the underlying vector space of n, and use it to define a bijection between conjugacy classes of trialitarian automorphisms and similarity classes of symmetric compositions. We isolate two types of symmetric compositions. Type I is related to octonion algebras; the fixed subgroups are of type G_{2}. When the characteristic is not 3, type II is related to central simple algebras of degree 3; the fixed subgroups are of type ^{1}A_{2} or ^{2}A_{2}. In characteristic 3, the fixed subgroups under trialitarian automorphisms of type II are not smooth.

Triality for simple Lie groups first appears in the paper [Car25] of É. Cartan, who already noticed that octonions can be used to explicitly define trialitarian automorphisms.^{1} The observation that symmetric compositions are particularly well suited for that purpose is due to M. Rost.

As far as we know a complete classification of trialitarian automorphisms of simple groups of type D_{4} had only been obtained over finite fields, in [GL83, (9.1)], and in [GLS98, Theorem (4.7.1)] for the algebraic closure of a finite field of characteristic different from 3. A summary of known results for Lie algebras can be found in [Knu09]. Like for duality, there is also a projective geometric version of triality, which is in fact older than Cartan’s triality and goes back to Study [Stu13]. A classification of geometric trialities, as well as of their groups of automorphisms, was done by Tits in [Tit59].

If not explicitly mentioned F denotes throughout the paper an arbitrary field.

2. Similarities of quadratic spaces

A quadratic space over F is a finite-dimensional vector space V over F with a quadratic form q: V → F. We always assume that q is nonsingular, in the sense that the polar bilinear form b_{q} defined by

b_{q}(x, y) = q(x + y) − q(x) − q(y) for x, y ∈ V

has radical {0}. We also assume throughout that dim V is even. Let ad_{q} denote the involution on End_{F} V such that

b_{q}(f(x), y) = b_{q}(x, ad_{q}(f)(y)) for all f ∈ End_{F} V and x, y ∈ V.

This involution is said to be adjoint to q. Let GO(q) be the F -algebraic group of similarities of (V, q), whose group of rational points GO(q)(F) consists of linear maps f: V → V for which there exists a scalar μ(f) ∈ F^{×}, called the multiplier of f, such that

1We refer to [KMRT98, pp. 510–511] and [SV00, §3.8] for historical comments on triality.

q(f(x)) = μ(f)q(x) for all x ∈ V.

The center of GO(q) is the multiplicative group G_{m}, whose rational points are viewed as homotheties. Let PGO(q) be the F -algebraic group of automorphisms of End_{F} V that commute with the adjoint involution ad_{q}. This group is identified with the quotient GO(q)/ G_{m}, acting on End_{F} V by inner automorphisms: for f ∈ GO(q)(F), we let [f] be the image of f in GO(q)(F)/ G_{m}(F) and identify [f] with

Int[f]: End_{F} V → End_{F} V, ϕ ↦ fϕf^{−1},

see [KMRT98, §23]. For simplicity, we write

GO(q) = GO(q)(F)  and  PGO(q) = PGO(q)(F) = GO(q)/F^{×}.

Let C(V, q) be the Clifford algebra of the quadratic space (V, q) and let C_{0}(V, q) be the even Clifford algebra. We let σ be the canonical involution of C(V, q), such that σ(x) = x for x ∈ V, and use the same notation for its restriction to C_{0}(V, q). Every similarity f ∈ GO(q) induces an automorphism C_{0}(f) of (C_{0}(V, q), σ) such that

(2.1)  C_{0}(f)(xy) = μ(f)^{−1}f(x)f(y)  for x, y ∈ V,

see [KMRT98, (13.1)]. This automorphism depends only on the image [f] = fF^{×} of f in PGO(q), and we shall use the notation C_{0}[f] for C_{0}(f). The similarity f is proper if C_{0}[f] fixes the center of C_{0}(V, q) and improper if it induces a nontrivial automorphism of the center of C_{0}(V, q) (see [KMRT98, (13.2)]). Proper similarities define an algebraic subgroup GO^{+}(q) in GO(q), and we let PGO^{+}(q) = GO^{+}(q)/ G_{m}, a subgroup of PGO(q). The groups GO^{+}(q) and PGO^{+}(q) are the connected components of the identity in GO(q) and PGO(q) respectively, see [KMRT98, §23.B]. Conjugation by an improper similarity is an outer automorphism of PGO^{+}(q), since the induced automorphism on the center of C_{0}(V, q) is nontrivial. As pointed out in the introduction, more outer automorphisms can be defined when the form q is the norm of an octonion algebra, i.e., a 3-fold Pfister form. In this case, we call the quadratic space a 3-fold Pfister quadratic space.

3. Symmetric compositions

Let (S, n) be a quadratic space of dimension 8 over F.

Definition 3.1. A symmetric composition on (S, n) is an F -bilinear map

⋆: S × S → S, (x, y) ↦ x ⋆ y for x, y ∈ S

subject to the following conditions:

(1) there exists λ_{⋆} ∈ F^{×}, called the multiplier of the symmetric composition ⋆, such that

n(x ⋆ y) = λ_{⋆}n(x)n(y) for all x, y ∈ S;
(2) for all x, y, z ∈ S,
b_{n}(x ⋆ y, z) = b_{n}(x, y ⋆ z).

A symmetric composition with multiplier λ_{⋆} = 1 is called normalized.

This definition of symmetric compositions is not identical to the one given in [KMRT98, §34], where (S, n) can a priori be a nonsingular quadratic space of arbitrary finite dimension (but in fact dim S = 1, 2, 4 or 8 by a theorem of Hurwitz, see [KMRT98, (33.28)]), and all the symmetric compositions are normalized.

The following lemma is an immediate consequence of the definition of a symmetric composition:

Lemma 3.2. Let ⋆ be a symmetric composition on (S, n) with multiplier λ_{⋆}.

(1) For any scalar ν ∈ F^{×}, ⋆ is a symmetric composition on the quadratic space (S, νn) with multiplier ν^{−1}λ_{⋆}.

(2) For any scalar ν ∈ F^{×}, the bilinear map ν ·⋆: (x, y) ↦ νx⋆y is a symmetric composition on the quadratic space (S, n) with multiplier ν^{2}λ_{⋆}.

Lemma 3.3. Under condition (1), condition (2) of the definition of a symmetric composition is equivalent to

x ⋆ (y ⋆ x) = λ_{⋆}n(x)y = (x ⋆ y) ⋆ x for all x, y ∈ S.

Proof. The claim follows by applying [KMRT98, (34.1)] to the (normalized) composition ⋆ on (S, λ_{⋆}n). □

Since the symmetric compositions do not change when the quadratic form n is scaled, we may and will always assume without loss of generality that n represents 1. It is then a 3-fold Pfister form, by [KMRT98, (33.18), (33.29)].

Example 3.4. Let (𝕆, n) be an octonion algebra with norm n, multiplication (x, y) ↦ x · y, identity 1, and conjugation x ↦ \overline{x}. The multiplication

x ⋆ y = \overline{x} · \overline{y}

defines a normalized symmetric composition on (𝕆, n), called the para-octonion composition (see for example [KMRT98, §34.A]). Observe that \overline{x ⋆ y} = \overline{y} ⋆ \overline{x} for all x, y ∈ 𝕆.

More examples—and a complete classification of symmetric compositions—are given in Section 9.

Definitions 3.5. Let (S, n) be a 3-fold Pfister quadratic space and let ⋆ and ⋄ be symmetric compositions on (S, n). A similarity f: ⋆ → ⋄ is an element f ∈ GO(n) such that

f(x ⋆ y) = f(x) ⋄ f(y) for all x, y ∈ S.

The multipliers of f, ⋆ and ⋄ are then related by λ_{⋆} = λ_{⋄}μ(f). Similarities with multiplier μ(f) = 1 are called isomorphisms. In particular, similarities between symmetric compositions with the same multiplier are isomorphisms.

The opposite of the symmetric composition ⋆ on (S, n) is the symmetric composition ⋆^{op} on (S, n) defined by

x ⋆^{op} y = y ⋆ x for x, y ∈ S.

The multiplier of ⋆^{op} is the same as the multiplier of ⋆.

Symmetric compositions ⋆ and ⋄ are said to be similar if there is a similarity ⋆ → ⋄; they are said to be antisimilar if ⋆^{op} and ⋄ are similar.

Proposition 3.6. Let n be a 3-fold quadratic Pfister form on a vector space S, and let ⋆ be a symmetric composition on (S, n). There exists up to isomorphism a unique normalized composition ⋄ on (S, n) similar to ⋆.

Proof. Let u ∈ S be such that n(u) = 1. Consider the maps ℓ_{u}^{⋆}, r_{u}^{⋆}: S → S defined by

ℓ_{u}^{⋆}(x) = u ⋆ x and r_{u}^{⋆}(x) = x ⋆ u for x ∈ S.

Define a new multiplication ⋄ on S by

x ⋄ y = λ_{⋆}^{−2}ℓ_{u}^{⋆}(r_{u}^{⋆}(x) ⋆ r_{u}^{⋆}(y)) for x, y ∈ S.

Condition (1) for a symmetric composition yields

n(ℓ_{u}^{⋆}(x)) = λ_{⋆}n(x) = n(r_{u}^{⋆}(x)) for all x ∈ S.

It is then easy to check that ⋄ is a normalized symmetric composition on (S, n). Moreover, by Lemma 3.3, we have

ℓ_{u}^{⋆} ∘ r_{u}^{⋆} = r_{u}^{⋆} ∘ ℓ_{u}^{⋆} = λ_{⋆} Id_{S} .

From the definition of ⋄, it follows that

λ_{⋆}^{−1}r_{u}^{⋆}(x ⋄ y) = (λ_{⋆}^{−1}r_{u}^{⋆}(x)) ⋆ (λ_{⋆}^{−1}r_{u}^{⋆}(y)) for all x, y ∈ S,

hence λ_{⋆}^{−1}r_{u}^{⋆}: ⋄ → ⋆ is a similarity. If ⋄′ is another normalized composition similar to ⋆, then ⋄ and ⋄′ are similar, hence isomorphic since they have the same multiplier.

□

4. From symmetric compositions to trialitarian automorphisms

Throughout this section, we fix a 3-fold Pfister quadratic space (S, n) and a symmetric composition ⋆ on (S, n) with multiplier λ_{⋆}. We show how to associate to this composition a trialitarian automorphism ρ_{⋆} of PGO^{+}(n) defined over F, i.e., an outer automorphism of order 3 in Aut(PGO^{+}(n))(F).

For each element x ∈ S we define linear maps ℓ_{x}^{⋆}, r_{x}^{⋆}: S → S by

ℓ_{x}^{⋆}(y) = x ⋆ y  and  r_{x}^{⋆}(y) = y ⋆ x  for y ∈ S.

Consider (\begin{matrix}0 & λ_{⋆}^{−1}r_{x}^{⋆} \\ ℓ_{x}^{⋆} & 0 \end{matrix}) ∈ M_{2}(End S) = End_{F} (S ⊕ S). By Lemma 3.3, we have

(\begin{matrix}0 & λ_{⋆}^{−1}r_{x}^{⋆} \\ ℓ_{x}^{⋆} & 0 \end{matrix})^{2} = n(x) · Id_{S⊕S} .

Therefore, by the universal property of Clifford algebras, the map

x ↦ (\begin{matrix}0 & λ_{⋆}^{−1}r_{x}^{⋆} \\ ℓ_{x}^{⋆} & 0 \end{matrix}) , x ∈ S

induces an F -algebra homomorphism

α^{⋆}: C(S, n) → End_{F} (S ⊕ S).

Proposition 4.1. The map α^{⋆} is an isomorphism of F -algebras

α^{⋆}: C(S, n) ⟶^{∼} End_{F} (S ⊕ S).

It restricts to an isomorphism of F -algebras with involution

α_{0}^{⋆}: (C_{0}(S, n), σ) ⟶^{∼} (End_{F} S, ad_{n}) × (End_{F} S, ad_{n})

such that for x, y ∈ S

α_{0}^{⋆}(x · y) = (λ_{⋆}^{−1}r_{x}^{⋆}ℓ_{y}^{⋆}, λ_{⋆}^{−1}ℓ_{x}^{⋆}r_{y}^{⋆}).

Proof. This is shown in [KMRT98, (35.1)]. For completeness, we reproduce the easy argument. The map α^{⋆} is injective since the Clifford algebra C(S, n) is simple. It restricts to an F -algebra embedding C_{0}(S, n) ↪ (End S) × (End S). Since dim S = 8, dimension count shows that this embedding is an isomorphism. Using Lemma 3.3, it is easy to check that the involution σ corresponds to ad_{n} × ad_{n} under α_{0}^{⋆}. □

For f, g, h ∈ GO(n), define the F -algebra automorphism Θ(g, h) of (End_{F} S) × (End_{F} S) by

Θ(g, h): (φ, ψ) ↦ (gψg^{−1}, hφh^{−1})

and consider the following diagrams:

D_{⋆}^{+}(f, g, h)  \xymatrix{C_{0}(S, n) \ar[r]^{α_{0}^{⋆}} \ar[d]_{C_{0}[f]} & (End_{F} S) × (End_{F} S) \ar[d]^{Int[g]×Int[h]} \\ C_{0}(S, n) \ar[r]^{α_{0}^{⋆}} & (End_{F} S) × (End_{F} S)}

and

D_{⋆}^{−}(f, g, h)  \xymatrix{C_{0}(S, n) \ar[r]^{α_{0}^{⋆}} \ar[d]_{C_{0}[f]} & (End_{F} S) × (End_{F} S) \ar[d]^{Θ(g,h)} \\ C_{0}(S, n) \ar[r]^{α_{0}^{⋆}} & (End_{F} S) × (End_{F} S).}

Lemma 4.2. Let ⋆ be a symmetric composition on (S, n). For f, g, h ∈ GO(n), the following statements are equivalent:

(a^{+}) there exists a scalar λ ∈ F^{×} such that

λf(x ⋆ y) = g(x) ⋆ h(y) for all x, y ∈ S;

(b^{+}) there exists a scalar μ ∈ F^{×} such that

μg(x ⋆ y) = h(x) ⋆ f(y) for all x, y ∈ S;

(c^{+}) there exists a scalar ν ∈ F^{×} such that

νh(x ⋆ y) = f(x) ⋆ g(y) for all x, y ∈ S;

(d^{+}) the diagram D_{⋆}^{+}(f, g, h) commutes; (e^{+}) the diagram D_{⋆}^{+}(g, h, f) commutes; (f^{+}) the diagram D_{⋆}^{+}(h, f, g) commutes.

When they hold, the scalars λ, μ, ν and the multipliers of f, g, h are related by

λμ = μ(h), μν = μ(f), λν = μ(g).

Moreover, the similarities f, g, and h are all proper in this case.

Likewise, for f, g, h ∈ GO(n) the following statements are equivalent:

(a^{−}) there exists a scalar λ ∈ F^{×} such that

λf(x ⋆ y) = h(y) ⋆ g(x) for all x, y ∈ S;

(b^{−}) there exists a scalar μ ∈ F^{×} such that

μg(x ⋆ y) = f(y) ⋆ h(x) for all x, y ∈ S;

(c^{−}) there exists a scalar ν ∈ F^{×} such that

νh(x ⋆ y) = g(y) ⋆ f(x) for all x, y ∈ S;

(d^{−}) the diagram D_{⋆}^{−}(f, h, g) commutes; (e^{−}) the diagram D_{⋆}^{−}(g, f, h) commutes; (f^{−}) the diagram D_{⋆}^{−}(h, g, f) commutes.

When they hold, the scalars λ, μ, ν and the multipliers of f, g, h are related by

λμ = μ(h), μν = μ(f), λμ = μ(g).

Moreover, the similarities f, g, and h are all improper in this case.

Proof. This is essentially proved in [KMRT98, (35.4)]. We give a proof for the reader’s convenience.

(a^{+})⇒(b^{+}) Multiplying each side of (a^{+}) on the left by h(y) and using Lemma 3.3, we obtain

λh(y) ⋆ f(x ⋆ y) = λ_{⋆}n(h(y))g(x) for x, y ∈ S.

If y is anisotropic, the map r_{y}^{⋆} is a bijection whose inverse is λ_{⋆}^{−1}n(y)^{−1}ℓ_{y}^{⋆}. Letting X = y and Y = x ⋆ y, we derive from the preceding equation

λh(X) ⋆ f(Y) = μ(h)g(X ⋆ Y) for X, Y ∈ S with Y anisotropic.

Since generic vectors in S are anisotropic, (b^{+}) follows with μ = μ(h)λ^{−1}. Similar arguments yield (b^{+})⇒(c^{+}) with ν = μ(f)μ^{−1} and (c^{+})⇒(a^{+}) with λ = μ(g)ν^{−1}.

Now, assume (a^{+}), (b^{+}), and (c^{+}) hold. From (b^{+}) and (c^{+}), and from μν = μ(f), we readily derive

g((y ⋆ z) ⋆ x) = μ^{−1}h(y ⋆ z) ⋆ f(x) = μ(f)^{−1}(f(y) ⋆ g(z)) ⋆ f(x)

and

h(x ⋆ (z ⋆ y)) = ν^{−1}f(x) ⋆ g(z ⋆ y) = μ(f)^{−1}f(x) ⋆ (h(z) ⋆ f(y))

for all x, y, z ∈ S. These equations mean that diagram D_{⋆}^{+}(f, g, h) commutes, hence (d^{+}) holds. Similar computations show that (e^{+}) and (f^{+}) hold.

Now, assume (d^{+}) holds. The map

x ↦ (\begin{matrix}0 & λ_{⋆}^{−1}r_{f(x)}^{⋆} \\ μ(f)^{−1}ℓ_{f(x)}^{⋆} & 0 \end{matrix})

also yields by the universal property of Clifford algebras an F -algebra isomorphism β: C(S, n) ⟶^{∼} End_{F} (S ⊕S). The automorphism β ∘(α^{⋆})^{−1} of End_{F} (S ⊕S) is inner by the Skolem–Noether theorem, and it preserves (End_{F} S) × (End_{F} S) diagonally embedded, hence

β ∘ (α^{⋆})^{−1} = Int (\begin{matrix}φ & 0 \\ 0 & ψ \end{matrix}) for some invertible φ, ψ ∈ End_{F} S.

It follows that for x ∈ S

φ ∘ r_{x}^{⋆} ∘ ψ^{−1} = r_{f(x)}^{⋆}  and  ψ ∘ ℓ_{x}^{⋆} ∘ φ^{−1} = μ(f)^{−1}ℓ_{f(x)}^{⋆},

which means that for x, y ∈ S

(4.3)  φ(y ⋆ x) = ψ(y) ⋆ f(x)  and  ψ(x ⋆ y) = μ(f)^{−1}f(x) ⋆ φ(y).

For x, y ∈ S we have

β(x · y) = (λ_{⋆}^{−1}μ(f)^{−1}r_{f(x)}^{⋆}ℓ_{f(x)}^{⋆}, λ_{⋆}^{−1}μ(f)^{−1}ℓ_{f(x)}^{⋆}r_{f(x)}^{⋆}) = α_{0}^{⋆} ∘ C_{0}[f](x · y),

hence (d^{+}) yields

Int (\begin{matrix}φ & 0 \\ 0 & ψ \end{matrix}) = Int (\begin{matrix}g & 0 \\ 0 & h \end{matrix}) .

It follows that [φ] = [g] and [ψ] = [h], hence (4.3) yields (c^{+}) and (b^{+}). Similarly, (e^{+})⇒(a^{+}), (c^{+}), and (f^{+})⇒(a^{+}), (b^{+}). Thus, all the statements (a^{+})–(f^{+}) are equivalent. When they hold, (d^{+}) shows that C_{0}[f] is the identity on the center of C_{0}(S, n), hence f is proper. Similarly, (e^{+}) shows that g is proper and (f^{+}) that h is proper.

The proof of the second part is similar. Details are left to the reader. □

Corollary 4.4. Let ⋆ be a symmetric composition on (S, n). Every automorphism ⋆ → ⋆ is a proper isometry.

Proof. For any automorphism f, condition (a^{+}) of Lemma 4.2 holds with λ = 1 and g = h = f. □

Theorem 4.5. Let ⋆ be a symmetric composition on (S, n).

(1) For any f ∈ GO^{+}(n), there exist g, h ∈ GO^{+}(n) such that the equivalent conditions (a^{+})–(f^{+}) in Lemma 4.2 hold.

(2) For any improper similarity f ∈ GO(n), there exist improper similarities g, h ∈ GO(n) such that the equivalent conditions (a^{−})–(f^{−}) in Lemma 4.2 hold.

In each case, the similarities g and h are uniquely determined up to multiplication by scalars, so [g] and [h] ∈ PGO(n) are uniquely determined.

Proof. (1) The automorphism C_{0}[f] of C_{0}(S, n) preserves the canonical involution σ and restricts to the identity on the center. Therefore, we may find g, h ∈ GO(n) such that the diagram D_{⋆}^{+}(f, h, g) commutes. The elements [g], [h] ∈ PGO(n) are uniquely determined by this condition (or by any other in the list (a^{+})–(f^{+}) from Lemma 4.2), and Lemma 4.2 shows that g and h are proper similarities.

(2) If f is improper, then C_{0}[f] restricts to the nontrivial automorphism of the center of C_{0}(S, n), hence it fits in a diagram D_{⋆}^{−}(f, h, g) for some similarities g, h ∈ GO(n), which are improper by Lemma 4.2. □

Note that scaling g and/or h we may change the scalars λ, μ, ν in conditions (a^{+})–(c^{+}) or (a^{−})–(c^{−}). We may for instance choose g and h so that λ = 1, or, as in [KMRT98, (35.4)], so that λ = μ(f)^{−1}, μ = μ(g)^{−1}, and ν = μ(h)^{−1}. In this case, μ(f)μ(g)μ(h) = 1.

In view of Theorem 4.5, we may define a map

ρ_{⋆}: PGO^{+}(n) → PGO^{+}(n)

by carrying any [f] ∈ PGO^{+}(n) to the unique [g] ∈ PGO^{+}(n) such that the diagram D_{⋆}^{+}(f, g, h) commutes for some h ∈ GO^{+}(n). Since ρ_{⋆} is defined in a functorial way, it actually defines a map of F -algebraic groups

ρ_{⋆}: PGO^{+}(n) → PGO^{+}(n).

Theorem 4.6. The map ρ_{⋆} is an outer automorphism of order 3 of PGO^{+}(n) and ρ_{⋆}^{−1} = ρ_{⋆^{op}}.

Proof. Since commutation of D_{⋆}^{+}(f, g, h) implies that D_{⋆}^{+}(g, h, f) and D_{⋆}^{+}(h, f, g) commute, we have

ρ_{⋆}[g] = [h]  and  ρ_{⋆}[h] = [f],

hence ρ_{⋆}^{3} = Id. Now, let f_{1}, f_{2} ∈ GO^{+}(n) and assume g_{1}, g_{2}, h_{1}, h_{2} ∈ GO^{+}(n) are such that D_{⋆}^{+}(f_{1}, g_{1}, h_{1}) and D_{⋆}^{+}(f_{2}, g_{2}, h_{2}) commute. By Lemma 4.2 we may find scalars λ_{1}, λ_{2} ∈ F^{×} such that for all x, y ∈ S

λ_{1}f_{1}(x ⋆ y) = g_{1}(x) ⋆ h_{1}(y)  and  λ_{2}f_{2}(x ⋆ y) = g_{2}(x) ⋆ h_{2}(y),

hence

λ_{1}λ_{2}f_{1}f_{2}(x ⋆ y) = λ_{1}f_{1}(g_{2}(x) ⋆ h_{2}(y)) = g_{1}g_{2}(x) ⋆ h_{1}h_{2}(y).

Therefore, ρ_{⋆}[f_{1}f_{2}] = [g_{1}g_{2}], showing that ρ_{⋆} is an automorphism. We refer to [KMRT98, (35.6)] for the fact that ρ_{⋆} is an outer algebraic group automorphism. (That it is outer also follows from the fact that ρ_{⋆} lifts to an automorphism of Spin(n) that is not the identity on the center, see (8.2)). □

The definition of ρ_{⋆} above depends on the isomorphism α_{0}^{⋆} through the diagram D_{⋆}^{+}(f, g, h). We next show that, conversely, the automorphism ρ_{⋆} determines α_{0}^{⋆}.

Proposition 4.7. Let ⋆ and ⋄ be symmetric compositions on (S, n). We have ρ_{⋆} = ρ_{⋄} if and only if α_{0}^{⋆} = α_{0}^{⋄}.

Proof. If α_{0}^{⋆} = α_{0}^{⋄}, then the diagrams D_{⋆}^{+}(f, g, h) and D_{⋄}^{+}(f, g, h) coincide for all f, g, h ∈ GO^{+}(n), hence ρ_{⋆} = ρ_{⋄}. Conversely, suppose that ρ_{⋆} = ρ_{⋄}. Then α_{0}^{⋄} ∘ (α_{0}^{⋆})^{−1} is an automorphism of (End S) × (End S) that commutes with the involution ad_{n} × ad_{n} and that makes the following diagram commute for any f ∈ GO^{+}(n)

(4.8)  \xymatrix{(End S) × (End S) \ar[r]^{α_{0}^{⋄}∘(α_{0}^{⋆})^{−1}} \ar[d]_{Int(ρ_{⋆}[f])×Int(ρ_{⋆}^{2}[f])} & (End S) × (End S) \ar[d]^{Int(ρ_{⋆}[f])×Int(ρ_{⋆}^{2}[f])} \\ (End S) × (End S) \ar[r]^{α_{0}^{⋄}∘(α_{0}^{⋆})^{−1}} & (End S) × (End S).}

Suppose first that α_{0}^{⋄} ∘ (α_{0}^{⋆})^{−1} exchanges the two factors, so there exist automorphisms θ_{1}, θ_{2} of (End S, ad_{n}) such that

α_{0}^{⋄} ∘ (α_{0}^{⋆})^{−1}(a, b) = (θ_{1}(b), θ_{2}(a)) for all a, b ∈ End S.

Automorphisms of (End S, ad_{n}) are inner automorphisms induced by similarities of (S, n), so we may find s_{1}, s_{2} ∈ GO(n) such that θ_{i} = Int[s_{i}] for i = 1, 2. By commutativity of diagram (4.8), we have

ρ_{⋆}[f] · [s_{1}] = [s_{1}] · ρ_{⋆}^{2}[f] and ρ_{⋆}^{2}[f] · [s_{2}] = [s_{2}] · ρ_{⋆}[f] for all f ∈ GO^{+}(n).

Thus, Int[s_{1}] ∘ ρ_{⋆}^{2} = ρ_{⋆}, hence Int[s_{1}] = ρ_{⋆}^{2} and therefore ρ_{⋆} = Int[s_{1}^{2}]. Since the square of any similarity is a proper similarity, it follows that ρ_{⋆} is an inner automorphism of PGO^{+}(n), a contradiction. Therefore, α_{0}^{⋄} ∘ (α_{0}^{⋆})^{−1} preserves the two factors, and we have

α_{0}^{⋄} ∘ (α_{0}^{⋆})^{−1} = Int[s_{1}] × Int[s_{2}]

for some s_{1}, s_{2} ∈ GO(n). Commutativity of diagram (4.8) yields

ρ_{⋆}[f] · [s_{1}] = [s_{1}] · ρ_{⋆}[f] and ρ_{⋆}^{2}[f] · [s_{2}] = [s_{2}] · ρ_{⋆}^{2}[f] for all f ∈ GO^{+}(n).

Therefore, [s_{1}] and [s_{2}] centralize PGO^{+}(n). It follows that s_{1} and s_{2} are proper similarities, since conjugation by an improper similarity is an outer automorphism of PGO^{+}(n). Since the center of PGO^{+}(n) is trivial, it follows that [s_{1}] = [s_{2}] = 1, hence α_{0}^{⋄} ∘ (α_{0}^{⋆})^{−1} = Id. □

5. The one-to-one correspondence

As in the preceding section, we fix a 3-fold Pfister quadratic space (S, n) over F. Our goal is to show that the map ⋆ ↦ ρ_{⋆} defines a one-to-one correspondence between symmetric compositions on (S, n) up to a scalar factor and trialitarian automorphisms of PGO^{+}(n) defined over F.

Recall that given a symmetric composition ⋆ on (S, n), linear operators ℓ_{x}^{⋆} and r_{x}^{⋆} on S are defined for any x ∈ S by

ℓ_{x}^{⋆}(y) = x ⋆ y  and  r_{x}^{⋆}(y) = y ⋆ x  for y ∈ S.

If x is anisotropic, condition (1) in the definition of symmetric compositions shows that ℓ_{x}^{⋆} and r_{x}^{⋆} are similarities of (S, n) with multiplier λ_{⋆}n(x). We shall see in Corollary 7.4 below that ℓ_{x}^{⋆} and r_{x}^{⋆} are improper similarities for every anisotropic vector x ∈ S. At this stage, we can at least prove:

Lemma 5.1. The following conditions are equivalent:

(a) there exists an anisotropic vector x ∈ S such that ℓ_{x}^{⋆} is an improper similarity;

(b) there exists an anisotropic vector x ∈ S such that r_{x}^{⋆} is an improper similarity;

(c) for every anisotropic vector x ∈ S, the similarity ℓ_{x}^{⋆} is improper;

(d) for every anisotropic vector x ∈ S, the similarity r_{x}^{⋆} is improper.

Proof. By Lemma 3.3 we have for all x ∈ S

r_{x}^{⋆} ∘ ℓ_{x}^{⋆} = λ_{⋆}n(x) · Id_{S},

and λ_{⋆}n(x) · Id_{S} is a proper similarity. Thus r_{x}^{⋆} and ℓ_{x}^{⋆} are either both proper or improper, proving (a) ⟺ (b) and (c) ⟺ (d). Since (c) ⇒ (a) is clear, it only remains to show (a) ⇒ (c). For this, we use a homotopy argument. Let x, y ∈ S be anisotropic vectors, and let t be an indeterminate over F. Consider the vector u(t) = x(1 − t) + yt ∈ S(t) = S_{F (t)}. It gives rise to a rational morphism

ϕ: 𝔸^{1} −−→ GO(S, n),

induced by t ↦ ℓ_{u(t)}^{⋆}, of the affine line over F to GO(S, n). Note that ϕ is defined at a if the vector u(a) = x(1 − a) + ya is anisotropic. Since GO(S, n) contains an improper similarity g of S, the variety of GO(S, n) is the disjoint union of two irreducible subvarieties GO^{+}(S, n) and g GO^{+}(S, n). Since 𝔸^{1} is irreducible the image of ϕ is contained either in GO^{+}(S, n) or in g GO^{+}(S, n). It follows immediately that the similarities ϕ(0) = x and ϕ(1) = y are both proper or both improper. □

Lemma 5.2. Let f, g ∈ GO(n) and let ⋆ be a symmetric composition on (S, n). The following conditions are equivalent:

(a) the map ⋄: S × S → S defined by

x ⋄ y = f(x) ⋆ g(y) for x, y ∈ S

is a symmetric composition on (S, n);

(b) for all x, y ∈ S,

f(x) ⋆ g(y) = μ(g)g^{−1}(x ⋆ f(y));

(c) f and g are proper similarities such that

ρ_{⋆}^{2}[f] · ρ_{⋆}[f] · [f] = 1 and [g] = ρ_{⋆}^{2}[f]^{−1} in PGO^{+}(n).

When they hold, the multiplier of ⋄ is λ_{⋄} = μ(f)μ(g)λ_{⋆} and

ρ_{⋄} = Int[f^{−1}] ∘ ρ_{⋆} = Int[g^{−1}] ∘ ρ_{⋆} ∘ Int[f].

Moreover, assuming (a)–(c) hold, the following conditions on [h] ∈ PGO^{+}(n) are equivalent:

(i) ρ_{⋄} = Int[h^{−1}] ∘ ρ_{⋆} ∘ Int[h],

(ii) [f] = ρ_{⋆}[h]^{−1} · [h] = [h] · ρ_{⋄}[h]^{−1}.

Proof. We first show (a) ⟺ (b). Since f and g are similarities and ⋆ is a symmetric composition, we have

(5.3)  n(f(x) ⋆ g(y)) = λ_{⋆}μ(f)μ(g)n(x)n(y)  for all x, y ∈ S,

hence the map ⋄ satisfies condition (1) in the definition of symmetric compositions. Therefore, (a) is equivalent to

b_{n}(f(x) ⋆ g(y), z) = b_{n}(x, f(y) ⋆ g(z)) for all x, y, z ∈ S.

Since ⋆ is a symmetric composition and g is a similarity, we may rewrite the right side as

b_{n}(x ⋆ f(y), g(z)) = μ(g)b_{n}(g^{−1}(x ⋆ f(y)), z).

Therefore, (a) is equivalent to

b_{n}(f(x) ⋆ g(y), z) = μ(g)b_{n}(g^{−1}(x ⋆ f(y)), z) for all x, y, z ∈ S.

Since n is nonsingular, this condition is also equivalent to (b).

Now, assume (b) holds. Fixing an anisotropic vector x ∈ S and considering each side of the equation in (b) as a function of y, we have

ℓ_{f(x)}^{⋆} ∘ g = μ(g)g^{−1} ∘ ℓ_{x}^{⋆} ∘ f.

By Lemma 5.1, the similarities ℓ_{f(x)}^{⋆} ∘ g and μ(g)g^{−1} ∘ ℓ_{x}^{⋆} are both proper or both improper, hence this equation shows that f is proper. Likewise, fixing an anisotropic vector y and considering each side of the equation in (b) as a function of x, we have

r_{g(y)}^{⋆} ∘ f = μ(g)g^{−1} ∘ r_{f(y)}^{⋆}.

Since by Lemma 5.1 the similarities r_{g(y)}^{⋆} and r_{f(y)}^{⋆} are either both proper or both improper, this equation shows that g is proper because f is proper.

By Lemma 4.2, condition (b) is equivalent to the commutativity of diagram D_{⋆}^{+}(g^{−1}, f, gf^{−1}), hence to

(5.4)  ρ_{⋆}[g^{−1}] = [f] and ρ_{⋆}^{2}[g^{−1}] = [gf^{−1}].

Since ρ_{⋆}^{3} = Id, it follows that [g] = ρ_{⋆}^{2}[f]^{−1} and ρ_{⋆}^{2}[f] · ρ_{⋆}[f] · [f] = 1. We have thus proved (b) ⇒ (c). Conversely, we readily deduce (5.4) from (c), hence the diagram D_{⋆}^{+}(g^{−1}, f, gf^{−1}) commutes. By Lemma 4.2, we may find λ, μ, ν ∈ F^{⋆} such that for all x, y ∈ S

(5.5)  λ g^{−1}(x ⋆ y) = f(x) ⋆ gf^{−1}(y),
(5.6)  μ f(x ⋆ y) = gf^{−1}(x) ⋆ g^{−1}(y),
(5.7)  ν gf^{−1}(x ⋆ y) = g^{−1}(x) ⋆ f(y),

and

λμ = μ(gf^{−1}), μν = μ(g^{−1}), λν = μ(f).

These last equations yield λ^{2}μν = μ(g). To obtain (b) from (5.5), it suffices to prove λ = μ(g), wich amounts to λμν = 1. For this, observe that

λμν x ⋆ y = λμν gf^{−1} ∘ f ∘ g^{−1}(x ⋆ y) for all x, y ∈ S.

Compute the right side using successively (5.5), (5.6), and (5.7):

λμν gf^{−1} ∘ f ∘ g^{−1}(x ⋆ y) = μν gf^{−1} ∘ f(f(x) ⋆ gf^{−1}(y))
= ν gf^{−1}(g(x) ⋆ f^{−1}(y))
= x ⋆ y.

Thus, λμν x ⋆ y = x ⋆ y for all x, y ∈ S, hence λμν = 1 and it follows that (c)⇒(b).

Now, assume (a), (b), and (c) hold. The equation λ_{⋄} = μ(f)μ(g)λ_{⋆} easily follows from (5.3). For φ ∈ GO^{+}(n) Theorem 4.5 yields φ′, φ′′ ∈ GO^{+}(n) such that

φ(x ⋆ y) = φ′(x) ⋆ φ′′(y) for all x, y ∈ S,

so ρ_{⋆}[φ] = [φ′]. Then

φ(x ⋄ y) = φ(f(x) ⋆ g(y)) = φ′f(x) ⋆ φ′′g(y) = f^{−1}φ′f(x) ⋄ g^{−1}φ′′g(y),

hence ρ_{⋄}[φ] = [f^{−1}φ′f]. It follows that ρ_{⋄} = Int[f^{−1}] ∘ ρ_{⋆}. From (c) it is easily derived that [f]^{−1} = [g^{−1}] · ρ_{⋆}[f], hence we also have ρ_{⋄} = Int[g^{−1}] ∘ ρ_{⋆} ∘ Int[f].

Finally, (i) holds if and only if ρ_{⋄} = Int([h]^{−1} · ρ_{⋆}[h]) ∘ ρ_{⋆}. This equation is equivalent to (ii) since ρ_{⋄} = Int[f^{−1}] ∘ ρ_{⋆} and the center of PGO^{+}(n) is trivial. □

Theorem 5.8. The assignment ⋆ ↦ ρ_{⋆} defines a one-to-one correspondence between symmetric compositions on (S, n) up to scalars and trialitarian automorphisms of PGO^{+}(n) defined over F.

Proof. We first show that the map is onto. Let τ be a trialitarian automorphism of PGO^{+}(n) over F and let ⋆ be a symmetric composition on (S, n), so ρ_{⋆} also is a trialitarian automorphism. In view of the exact sequence (1.1), we have either τ ≡ ρ_{⋆} or τ ≡ ρ_{⋆}^{−1} mod Int(PGO^{+}(n)). Substituting ⋆^{op} for ⋆ if necessary, we may assume τ ≡ ρ_{⋆} mod Int(PGO^{+}(n)), hence there exists f ∈ GO^{+}(n) such that

τ = Int[f^{−1}] ∘ ρ_{⋆}.

Since τ^{3} = ρ_{⋆}^{3} = Id, we must have [f]^{−1} · ρ_{⋆}[f]^{−1} · ρ_{⋆}^{2}[f]^{−1} = 1. Let g ∈ GO^{+}(n) be such that [g] = ρ_{⋆}^{2}[f]^{−1}. Lemma 5.2 then shows that τ = ρ_{⋄} for the symmetric composition ⋄ defined by

x ⋄ y = f(x) ⋆ g(y) for x, y ∈ S.

Now, let ⋆ and ⋄ be two symmetric compositions on (S, n). Suppose ⋄ is a scalar multiple of ⋆, say there exists μ ∈ F^{×} such that x ⋄ y = μx ⋆ y for all x, y ∈ S. Then λ_{⋄} = μ^{2}λ_{⋆}, and for all x ∈ S we have ℓ_{x}^{⋄} = μℓ_{x}^{⋆} and r_{x}^{⋄} = μr_{x}^{⋆}. Therefore, α_{0}^{⋄} = α_{0}^{⋆}, hence ρ_{⋄} = ρ_{⋆} by Proposition 4.7. Thus, symmetric compositions that are scalar multiples of each other define the same trialitarian automorphism, and it only remains to show the converse: if ⋆ and ⋄ are symmetric compositions such that ρ_{⋆} = ρ_{⋄}, then ⋆ and ⋄ are multiples of each other. By Proposition 4.7, the hypothesis ρ_{⋆} = ρ_{⋄} implies α_{0}^{⋆} = α_{0}^{⋄}, hence

λ_{⋆}^{−1}ℓ_{x}^{⋆}r_{y}^{⋆} = λ_{⋄}^{−1}ℓ_{x}^{⋄}r_{y}^{⋄} for all x, y ∈ S.

Fix an anisotropic vector x ∈ S and let φ = (ℓ_{x}^{⋄})^{−1}ℓ_{x}^{⋆}, so

λ_{⋄}λ_{⋆}^{−1}φr_{y}^{⋆} = r_{y}^{⋄} for all y ∈ S.

This equation means that

(5.9)  z ⋄ y = λ_{⋄}λ_{⋆}^{−1}φ(z ⋆ y)  for all y, z ∈ S.

The map φ is a similarity since ℓ_{x}^{⋄} and ℓ_{x}^{⋆} are similarities. Its multiplier is

μ(φ) = μ(ℓ_{x}^{⋄})^{−1}μ(ℓ_{x}^{⋆}) = λ_{⋄}^{−1}λ_{⋆}.

Suppose first φ is improper. Theorem 4.5 then yields improper similarities φ′, φ′′ such that

φ(z ⋆ y) = μ(φ)φ′(y) ⋆ φ′′(z) for all y, z ∈ S,

hence

z ⋄ y = φ′′(z) ⋆^{op} φ′(y) for all y, z ∈ S.

Lemma 5.2 yields a contradiction since φ′ and φ′′ are improper. Therefore, φ is proper and Theorem 4.5 yields φ′, φ′′ ∈ GO^{+}(n) such that

φ(z ⋆ y) = μ(φ)φ′(z) ⋆ φ′′(y) for all y, z ∈ S,

hence

z ⋄ y = φ′(z) ⋆ φ′′(y) for all y, z ∈ S.

By Lemma 5.2 it follows that ρ_{⋄} = Int[φ′^{−1}] ∘ ρ_{⋆}. Since by hypothesis ρ_{⋆} = ρ_{⋄}, this equation implies [φ′] = 1. But [φ′] = ρ_{⋆}[φ], so also [φ] = 1. Equation (5.9) then shows that ⋄ and ⋆ are scalar multiples of each other. □

6. Classification of conjugacy classes of trialitarian automorphisms

We next show that under the one-to-one correspondence of Theorem 5.8 similarity of symmetric compositions corresponds to conjugacy of trialitarian automorphisms.

Proposition 6.1. Let h: ⋄ → ⋆ be a similarity of symmetric compositions on (S, n). Then

(6.2)  ρ_{⋆} = Int[h] ∘ ρ_{⋄} ∘ Int[h]^{−1}.

Conversely, if (6.2) holds for some h ∈ GO(n), then some scalar multiple of h is a similarity ⋄ → ⋆.

Proof. Suppose first h: ⋄ → ⋆ is a similarity of symmetric compositions, so

(6.3)  h(x ⋄ y) = h(x) ⋆ h(y)  for all x, y ∈ S.

Let φ ∈ GO^{+}(n). Theorem 4.5 yields φ′, φ′′ ∈ GO^{+}(n) such that

φ(x ⋆ y) = φ′(x) ⋆ φ′′(y) for all x, y ∈ S,

and ρ_{⋆}[φ] = [φ′] by definition. Applying φ to each side of (6.3), we find

φh(x ⋄ y) = φ′h(x) ⋆ φ′′h(y).

The right side is

hh^{−1}φ′h(x) ⋆ hh^{−1}φ′′h(y) = h(h^{−1}φ′h(x) ⋄ h^{−1}φ′′h(y)),

hence

h^{−1}φh(x ⋄ y) = h^{−1}φ′h(x) ⋄ h^{−1}φ′′h(y) for all x, y ∈ S.

Therefore, ρ_{⋄}([h^{−1}φh]) = [h^{−1}] · ρ_{⋆}[φ] · [h], proving (6.2).

Conversely, assume (6.2) holds, and set

x ▷ y = h^{−1}(h(x) ⋆ h(y)) for x, y ∈ S,

so ▷ is a symmetric composition similar to ⋆ under h. The arguments above show that

ρ_{▷} = Int[h^{−1}] ∘ ρ_{⋆} ∘ Int[h] = ρ_{⋄},

hence Theorem 5.8 implies that ⋄ is a multiple of ▷. If λ ∈ F^{×} is such that x ▷ y = λ x ⋄ y for all x, y ∈ S, then λ h(x ⋄ y) = h(x) ⋆ h(y), hence

λ^{−1}h(x ⋄ y) = λ^{−1}h(x) ⋆ λ^{−1}h(y) for all x, y ∈ S.

This equation shows that λ^{−1}h: ⋄ → ⋆ is a similarity. □

For the following statement, recall from Proposition 3.6 that every symmetric composition is similar to a normalized symmetric composition, which is unique up to isomorphism.

Theorem 6.4. Let ⋆ and ⋄ be symmetric compositions on (S, n), and let \overline{⋆}, \overline{⋄} be normalized symmetric compositions similar to ⋆ and to ⋄ respectively. The following conditions are equivalent:

(a) ⋆ and ⋄ are similar;

(b) \overline{⋆} and \overline{⋄} are isomorphic;

(c) ρ_{⋆} and ρ_{⋄} are conjugate in Aut(PGO^{+}(n)) over F;

(d) ρ\overline{⋆} and ρ\overline{⋄} are conjugate in Aut(PGO^{+}(n)) over F.

Proof. It is clear from the definition of \overline{⋆} and \overline{⋄} that (a) ⟺ (b). Since ⋆ and \overline{⋆} are similar, Proposition 6.1 shows that ρ_{⋆} and ρ\overline{⋆} are conjugate in Aut(PGO^{+}(n)) over F. Similarly, ρ_{⋄} and ρ\overline{⋄} are conjugate, hence it is clear that (c) ⟺ (d). Proposition 6.1 also yields (a)⇒(c), so it only remains to show (c)⇒(a).

Suppose ϕ is an automorphism of PGO^{+}(n) defined over F such that

ρ_{⋆} = ϕ ∘ ρ_{⋄} ∘ ϕ^{−1}.

Let f ∈ GO(n) be an improper similitude. The restriction of Int[f] to PGO^{+}(n) is an outer automorphism whose square is inner, hence each coset of Aut(PGO^{+}(n)) modulo Int(PGO^{+}(n)) is represented over F by an element from the set

{Id, ρ_{⋄}, ρ_{⋄}^{2}, Int[f], Int[f] ∘ ρ_{⋄}, Int[f] ∘ ρ_{⋄}^{2}}.

Since ϕ ∘ ρ_{⋄} ∘ ϕ^{−1} does not change when ϕ is multiplied on the right by ρ_{⋄} or ρ_{⋄}^{−1}, we may assume the coset of ϕ is represented by Id or by Int[f], hence ϕ = Int[h] for some h ∈ GO(n). It follows from Proposition 6.1 that ⋆ and ⋄ are similar. □

Theorem 6.4 shows that the correspondence ⋆ ↦ ρ_{⋆} induces a one-to-one correspondence between similarity classes of symmetric compositions, or isomorphism classes of normalized symmetric compositions, and conjugacy classes of trialitarian automorphisms over F in Aut(PGO^{+}(n)).

Corollary 6.5. (1) Two symmetric compositions ⋆ and ⋄ on (S, n) are similar or antisimilar if and only if the subgroups generated by ρ_{⋆} and ρ_{⋄} in Aut(PGO^{+}(n)) are conjugate.

(2) Two normalized symmetric compositions ⋆ and ⋄ on (S, n) are isomorphic or anti-isomorphic if and only if the subgroups generated by ρ_{⋆} and ρ_{⋄} in Aut(PGO^{+}(n)) are conjugate.

Proof. The subgroups generated by ρ_{⋆} and ρ_{⋄} are conjugate if and only if ρ_{⋆} is conjugate to ρ_{⋄} or to ρ_{⋄}^{−1}. Since ρ_{⋄}^{−1} = ρ_{⋄^{op}} by Theorem 4.6, the corollary follows from Theorem 6.4. □

To complete this section, we discuss fixed points of trialitarian automorphisms, which will be used to classify trialitarian automorphisms in Section 9. For any symmetric composition ⋆ on (S, n), we let Aut(⋆) be the F -algebraic group of automorphisms of ⋆, whose group of F -rational points Aut(⋆) consists of the isomorphisms f: ⋆ → ⋆. Corollary 4.4 shows that Aut(⋆) ⊂ GO^{+}(n).

Theorem 6.6. Let ⋆ be a symmetric composition on (S, n). The canonical map ϕ: GO^{+}(n) → PGO^{+}(n) induces an isomorphism from Aut(⋆) to the subgroup PGO^{+}(n)^{ρ_{⋆}} of PGO^{+}(n) fixed under the trialitarian automorphism ρ_{⋆}.

Proof. If f ∈ Aut(⋆), then ρ_{⋆}[f] = [f], hence ϕ gives rise to a canonical map

ψ: Aut(⋆) → PGO^{+}(n)^{ρ_{⋆}}.

To prove that ψ is an isomorphism it suffices to show that Ker ψ = 1 and that ψ is a quotient map (see [Wat79, Corollary 15.4]). The kernel of ψ represents the functor which takes a commutative F -algebra A to Ker[Aut(⋆)(A) → PGO^{+}(n)^{ρ_{⋆}}(A)]. Let a be in this kernel. Note that Ker ψ ⊂ Ker ϕ = G_{m}. Hence a is a homothety r Id_{S_{A}} for some r ∈ A^{×}. But such a map is in Aut(⋆)(A) if and only if r = 1. Thus Ker ϕ represents the trivial functor and so Ker ϕ = 1. To see that ψ is a quotient map we need to show that for every commutative F -algebra A and every g ∈ PGO^{+}(n)^{ρ_{⋆}}(A) there exists a faithfully flat extension A → B and an element f ∈ Aut(⋆)(B) such that ψ(f) = g (see [Wat79, Theorem 15.5]). Since ϕ is a quotient map we can find an F -algebra B and an element f_{0} ∈ GO^{+}(n)(B) such that ϕ(f_{0}) = g. Since g is fixed by ρ_{⋆} there are scalars α, β ∈ B^{×} such that

f_{0}(x ⋆ y) = αf_{0}(x) ⋆ βf_{0}(y) for all x, y ∈ S_{B}.

Let f = αβf_{0}. Then f is an automorphism of ⋆ and ψ(f) = ϕ(f_{0}) = g. □

7. The twist of a symmetric composition

As in the preceding sections, we fix a 3-fold Pfister quadratic space (S, n) over F. If ⋆ and ⋄ are symmetric compositions on (S, n), then the trialitarian automorphisms ρ_{⋆} and ρ_{⋄} are related by inner automorphisms of PGO^{+}(n): in view of the exact sequence (1.1), we have either

ρ_{⋆} ≡ ρ_{⋄} mod Int(PGO^{+}(n)) or ρ_{⋆} ≡ ρ_{⋄}^{−1} mod Int(PGO^{+}(n)).

We use this observation to relate the symmetric compositions ⋆ and ⋄.

Proposition 7.1. Let ⋆ and ⋄ be two symmetric compositions on (S, n). If ρ_{⋆} ≡ ρ_{⋄} mod Int(PGO^{+}(n)), then there exist f, g ∈ GO^{+}(n) such that

(7.2)  x ⋄ y = f(x) ⋆ g(y)  for all x, y ∈ S.

If ρ_{⋆} ≡ ρ_{⋄}^{−1} mod Int(PGO^{+}(n)), then there exist f, g ∈ GO^{+}(n) such that

x ⋄ y = g(y) ⋆ f(x) for all x, y ∈ S.

Proof. Suppose first ρ_{⋄} = Int[f^{−1}] ∘ ρ_{⋆} for some f ∈ GO^{+}(n). Since ρ_{⋄}^{3} = Id, we have ρ_{⋆}^{2}[f] · ρ_{⋆}[f] · [f] = 1. Let g ∈ GO^{+}(n) be such that [g] = ρ_{⋆}^{2}[f^{−1}], and define

x ▷ y = f(x) ⋆ g(y) for x, y ∈ S.

By Lemma 5.2, ▷ is a symmetric composition on (S, n), and

ρ_{▷} = Int[f^{−1}] ∘ ρ_{⋆} = ρ_{⋄}.

Therefore, Theorem 5.8 shows that ▷ is a scalar multiple of ⋄. Scaling g, we may assume (7.2) holds.

If ρ_{⋆} ≡ ρ_{⋄}^{−1} mod Int(PGO^{+}(n)), then ρ_{⋆^{op}} ≡ ρ_{⋄} mod Int(PGO^{+}(n)) since ρ_{⋆^{op}} = ρ_{⋆}^{−1} by Theorem 4.6. The first part of the proof yields f, g ∈ GO^{+}(n) such that

x ⋄ y = f(x) ⋆^{op} g(y) = g(y) ⋆ f(x) for x, y ∈ S.
□

When the symmetric composition ⋄ is given by (7.2), then we must have [g] = ρ_{⋆}^{2}[f^{−1}], by Lemma 5.2. The map g is therefore uniquely determined by f up to a scalar factor, and we say ⋄ is a twist of ⋆ through the similarity f. Thus, by Proposition 7.1, given a symmetric composition ⋆ on (S, n), every symmetric composition on (S, n) is a twist of ⋆ or ⋆^{op}.

Twisting has its origin in Petersson [Pet69], where the following special case is considered: suppose f: ⋆ → ⋆ is a similarity (i.e., f is an automorphism of ⋆), and f^{3} = Id_{S}. Since f is an automorphism, we have f ∈ GO^{+}(n) and ρ_{⋆}[f] = [f]: see Theorem 6.6. Since f^{3} = Id_{S}, we have ρ_{⋆}^{2}[f] · ρ_{⋆}[f] · [f] = 1, hence we may choose g = f^{−1} in the discussion above. By Lemma 5.2, the product

(7.3)  x ⋆_{f} y = f(x) ⋆ f^{−1}(y)  for x, y ∈ S

defines a symmetric composition.

The idea of twisting was further developed in [Eld00].

Corollary 7.4. For every symmetric composition ⋄ on (S, n) and every anisotropic vector x ∈ S, the similarities ℓ_{x}^{⋄} and r_{x}^{⋄} are improper.

Proof. Let ⋆ be a para-octonion composition on (S, n) (see Example 3.4). The similarities ℓ_{1}^{⋆} and r_{1}^{⋆}, where 1 is the identity of the octonion algebra, coincide with the conjugation map, which is an improper isometry. Therefore, by Lemma 5.1, the similarities ℓ_{x}^{⋆} and r_{x}^{⋆} are improper for every anisotropic vector x ∈ S. Propositiom 7.1 shows that there exist f, g ∈ GO^{+}(n) such that either

x ⋄ y = f(x) ⋆ g(y) for all x, y ∈ S or x ⋄ y = g(y) ⋆ f(x) for all x, y ∈ S.

Therefore, for every anisotropic vector x ∈ S we have either ℓ_{x}^{⋄} = ℓ_{f(x)}^{⋆} ∘ g and r_{x}^{⋄} = r_{g(x)}^{⋆} ∘ f, or ℓ_{x}^{⋄} = r_{f(x)}^{⋆} ∘ g and r_{x}^{⋄} = ℓ_{g(x)}^{⋆} ∘ f. The similarities ℓ_{x}^{⋄} and r_{x}^{⋄} are therefore improper. □

8. Spin groups and symmetric compositions

Let ⋆ be a normalized symmetric composition on a 3-fold Pfister quadratic space (S, n). We briefly point out in this section how the results of the preceding sections can be modified to apply to the group Spin(n) instead of PGO^{+}(n).

Recall that the F -rational points of the F -algebraic group Spin(n) are given by

Spin(n) = {c ∈ C_{0}(S, n)^{×} | cSc^{−1} ⊂ S and cσ(c) = 1}

(see for example [KMRT98, 35.C.]). The isomorphism α_{0}^{⋆} introduced in Proposition 4.1 can be used to give a convenient description of Spin(n): for c ∈ Spin(n), define f, f_{1}, f_{2} ∈ End_{F} S as follows:

f(x) = cxc^{−1} for x ∈ S, α_{0}^{⋆}(c) = (f_{1}, f_{2}).

The map f is a proper isometry. Similarly, since cσ(c) = 1 and α_{0}^{⋆} is an isomorphism of algebras with involution, f_{1} and f_{2} are isometries. We have C_{0}[f] = Int(c)|_{C_{0}(S,n)}, hence the diagram D_{⋆}^{+}(f, f_{1}, f_{2}) commutes. Therefore, f_{1} and f_{2} are proper isometries. Let O^{+}(n) be the group of proper isometries of (S, n). The following result is shown in [KMRT98, (35.7)] (assuming char F ≠ 2):

Proposition 8.1. The map c ↦ (f, f_{1}, f_{2})_{⋆} defines an isomorphism

Spin(n) ⟶^{∼} {(f, f_{1}, f_{2})_{⋆} | f_{i} ∈ O^{+}(n)(F), f(x ⋆ y) = f_{1}(x) ⋆ f_{2}(y), x, y ∈ S}.

Moreover any of the three relations

\begin{aligned}f(x ⋆ y) & = & f_{1}(x) ⋆ f_{2}(y) \\ f_{1}(x ⋆ y) & = & f_{2}(x) ⋆ f(y) \\ f_{2}(x ⋆ y) & = & f(x) ⋆ f_{1}(y) \end{aligned}

implies the two others.

Observe that the representation of elements of Spin(n) as triples of elements of O^{+}(n) depends on the choice of the composition ⋆, hence the notation (f, f_{1}, f_{2})_{⋆}.

We have an obvious trialitarian automorphism ρ̂_{⋆} of Spin(n) defined over F by

ρ̂_{⋆}: (f, f_{1}, f_{2})_{⋆} ↦ (f_{1}, f_{2}, f)_{⋆}.

The center Z of Spin(n) is the scheme μ_{2}^{2} with μ_{2}(F) = ±1. Viewing Z(F) as kernel of the multiplication map

μ_{2}^{3}(F) → μ_{2}(F), (ε_{1}, ε_{2}, ε_{3}) ↦ ε_{1}ε_{2}ε_{3},

the group scheme Z admits a natural 𝔖_{3}-action. The exact sequence of schemes

(8.2)  1 → Z → Spin(n) → PGO^{+}(n) → 1,

where the morphism Spin(n) → PGO^{+}(n) is induced by (f, f_{1}, f_{2})_{⋆} ↦ [f], is 𝔖_{3}- equivariant (see [KMRT98, (35.13)]). Thus the trialitarian action on Spin(n) is a lift of the trialitarian action on PGO^{+}(n).

Most of the results on trialitarian actions on PGO^{+}(n) hold for Spin(n). Moreover, in a similar way we have a natural action of 𝔖_{3} on Spin(n). Let H be the semidirect product of Spin(n) and 𝔖_{3}.

Theorem 8.3. Two normalized symmetric compositions ⋆ and ⋄ on (S, n) are isomorphic if and only if the trialitarian automorphisms ρ̂_{⋆} and ρ̂_{⋄} are conjugate over F in H.

One of the main steps in the proof for PGO^{+}(n) was to show that any trialitarian automorphism is induced by a symmetric composition. We describe this step for Spin(n).

Proposition 8.4. Every trialitarian automorphism of Spin(n) over F in H has the form ρ̂_{⋄} for some normalized symmetric composition ⋄.

Proof. We show how to modify the proof of the corresponding statement in Theorem 5.8 for PGO^{+}(n). Let τ be a trialitarian automorphism of Spin(n) over F, let ⋆ be a symmetric composition on (S, n) and let ρ̂_{⋆} be the associated trialitarian automorphism. In view of the exact sequence (1.1), we have either τ ≡ ρ̂_{⋆} or τ ≡ ρ_{⋆}^{−1} mod Spin(n). Substituting ⋆^{op} for ⋆ if necessary, we may assume τ ≡ ρ̂_{⋆} mod Spin(n), hence there exists (h, h_{1}, h_{2})_{⋆} ∈ Spin(n) such that

τ = Int ((h, h_{1}, h_{2})_{⋆}^{−1}) ∘ ρ̂_{⋆},

which means that for all (f, f_{1}, f_{2})_{⋆} ∈ Spin(n),

τ((f, f_{1}, f_{2})_{⋆}) = (h^{−1}f_{1}h, h_{1}^{−1}f_{2}h_{1}, h_{2}^{−1}fh_{2})_{⋆}.

It follows from τ^{3} = ρ_{⋆}^{3} = Id that h_{2}h_{1}h = Id_{S}. Let

x ⋄ y = h(x) ⋆ h_{2}^{−1}(y) for x, y ∈ S.

One deduces easily from h_{2}h_{1}h = Id_{S} that (x⋄y)⋄x = x⋄(y⋄x) = n(x)y for x, y ∈ S. Thus ⋄ is a normalized symmetric composition on (S, n). For (f, f_{1}, f_{2})_{⋆} ∈ Spin(n) and x, y ∈ S we have

(8.5)  \begin{aligned}f(x ⋄ y) & = & f(h(x) ⋆ h_{2}^{−1}(y)) \\ & = & f_{1}h(x) ⋆ f_{2}h_{2}^{−1}(y) \\ & = & h^{−1}f_{1}h(x) ⋄ h_{2}f_{2}h_{2}^{−1}(y) \end{aligned}

Let

Spin(n) = {(f, f′_{1}, f′_{2})_{⋄} | f, f′_{1}, f′_{2} ∈ O^{+}(n), f(x ⋄ y) = f′_{1}(x) ⋆ f′_{2}(y), x, y ∈ S}

be the presentation of Spin(n) using the symmetric composition ⋄. It follows from (8.5) that

f′_{1} = h^{−1}f_{1}h and f′_{2} = h_{2}^{−1}f_{2}h_{2},

so that the passage from the presentation of Spin(n) using ⋆ to the presentation of Spin(n) using ⋄ is described as

(8.6)  (f, f_{1}, f_{2})_{⋆} ↦ (f, h^{−1}f_{1}h, h_{2}f_{2}h_{2}^{−1})_{⋄}.

Since ρ̂_{⋄}(f, f′_{1}, f′_{2})_{⋄} = (f′_{1}, f′_{2}, f)_{⋄}, we get, using (8.6),

ρ̂_{⋄}((f, f_{1}, f_{2})_{⋆}) = (h^{−1}f_{1}h, hh_{2}f_{2}h_{2}^{−1}h^{−1}, h_{2}^{−1}fh_{2})_{⋆}

so that the relation h_{2}h_{1}h = Id_{S} implies τ = ρ̂_{⋄}. □

9. Classification of symmetric compositions

It follows from Theorem 6.4 that the classification of symmetric compositions up to similarity, the classification of normalized symmetric compositions up to isomorphism, and the classification of trialitarian automorphisms up to conjugation are essentially equivalent. In this section we recall the classification of normalized symmetric compositions of dimension 8 over arbitrary fields.

We first consider symmetric compositions over algebraically closed fields. The norm form n is then hyperbolic. Octonion algebras over such fields are split and it is convenient to choose as a model the Zorn algebra, which is defined in arbitrary characteristic.

The Zorn algebra. We denote by • the usual scalar product on F^{3} = F × F × F, and by × the vector product: for a = (a_{1}, a_{2}, a_{3}) and b = (b_{1}, b_{2}, b_{3}) ∈ F^{3},

a • b = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}  and  a × b = (a_{2}b_{3} − a_{3}b_{2}, a_{3}b_{1} − a_{1}b_{3}, a_{1}b_{2} − a_{2}b_{1}).

The Zorn algebra is the set of matrices

ℨ = \{(\begin{matrix}α & a \\ b & β \end{matrix})| α, β ∈ F, a, b ∈ F^{3}\}

with the product

(\begin{matrix}α & a \\ b & β \end{matrix}) · (\begin{matrix}γ & c \\ d & δ \end{matrix}) = (\begin{matrix}αγ + a • d & αc + δa − b × d \\ γb + βd + a × c & βδ + b • c \end{matrix}) ,

the norm

n (\begin{matrix}α & a \\ b & β \end{matrix}) = αβ − a • b,

and the conjugation

\overline{(\begin{matrix}α & a \\ b & β \end{matrix})} = (\begin{matrix}β & −a \\ −b & α \end{matrix}) ,

which is such that ξ · \overline{ξ} = \overline{ξ} · ξ = n(ξ) for all ξ ∈ ℨ (see [Zor30, p. 144]).

In view of Example 3.4 the product

(\begin{matrix}α & a \\ b & β \end{matrix})∗(\begin{matrix}γ & c \\ d & δ \end{matrix}) = \overline{(\begin{matrix}α & a \\ b & β \end{matrix})}·\overline{(\begin{matrix}γ & c \\ d & δ \end{matrix})} = (\begin{matrix}βδ + a • d & −βc − γa − b × d \\ −δb − αd + a × c & αγ + b • c \end{matrix}) ,

defines a symmetric composition on the quadratic space (ℨ, n). We call ⋆ the para-Zorn composition.

We use the technique of Section 7 to twist the para-Zorn composition. Let p: F^{3} → F^{3} be the map (x_{1}, x_{2}, x_{3}) ↦ (x_{3}, x_{1}, x_{2}), and let

π: ℨ → ℨ, (\begin{matrix}α & a \\ b & β \end{matrix}) = (\begin{matrix}α & p(a) \\ p(b) & β \end{matrix}) .

The map π is an automorphism of ⋆ of order 3, so we may consider the twisted composition ⋆_{π} as in (7.3), which we call the split Petersson symmetric composition. (The algebra (ℨ, ⋆_{π}) is also known as the split pseudo-octonion algebra.)

Theorem 9.1. Over an algebraically closed field, there are exactly two symmetric compositions up to isomorphism: the para-Zorn composition and the split Petersson composition.

Proof. The claim is a result of Petersson [Pet69, Satz 2.7] if the field has characteristic different from 2 and 3, and is due to Elduque-Pérez [EP96] in arbitrary characteristic. □

Over arbitrary fields we consider two kinds of symmetric compositions, which we call type I and type II. Symmetric compositions of type I are forms of the para-Zorn composition, i.e., ⋆ is of type I if and only if ⋆ is isomorphic to the para-Zorn composition after scalar extension to an algebraic closure. Similarly, we say that ⋆ is of type II if it is a form of the split Petersson composition. Thus, Theorem 9.1 shows that every symmetric composition is either of type I or of type II.

Type I. The classification of symmetric compositions of type I is particularly simple.

Theorem 9.2. Let (S, n) be a 3-fold Pfister quadratic space over an arbitrary field F. Up to isomorphism, there is a unique normalized symmetric composition of type I on (S, n), given by the para-octonion composition.

Proof. Forms of the split para-octonion algebra are para-octonion algebras (see for example [KMRT98, §34 A]), hence symmetric compositions of type I are paraoctonion compositions as defined in Example 3.4. Isomorphisms of octonion algebras are isomorphisms of the corresponding para-octonion algebras, and octonion algebras with isometric norms are isomorphic (see for example [KMRT98, (33.19)]). □

The automorphism group of a symmetric composition of type I is the automorphism group of the corresponding octonion algebra; it is a simple algebraic group of type G_{2}. Accordingly, symmetric compositions of type I are also called symmetric compositions of type G_{2}.

Type II. The classification of symmetric compositions of type II has a completely different flavor in characteristic 3. We first discuss the case where the characteristic is different from 3, and distinguish two subcases, depending on whether the base field contains a primitive cube root of unity or not.

Suppose first F is a field of characteristic different from 3 containing a primitive cube root of unity ω. Let A be a central simple F -algebra of degree 3. For the reduced characteristic polynomial of a ∈ A, we use the notation

X^{3} − Trd(a)X^{2} + Srd(a)X − Nrd(a) 1,

so Trd is the reduced trace map on A, Srd is the reduced quadratic trace map, and Nrd is the reduced norm. Let A^{0} ⊂ A be the kernel of Trd. We define a multiplication ⋆ on A^{0} by

(9.3)  x ⋆ y = \frac{yx − ωxy}{1 − ω} − \frac{1}{3} Trd(xy) 1

and a quadratic form n by

(9.4)  n(x) = −\frac{1}{3} Srd(x).

The following result can be found for instance in [KMRT98, (34.19), (34.25)]:

Proposition 9.5. The quadratic space (A^{0}, n) is hyperbolic, and ⋆ is a normalized symmetric composition on (A^{0}, n).

When the algebra A is split, the composition ⋆ is isomorphic to the split Petersson composition. Symmetric compositions as in Proposition 9.5 are called Okubo compositions. The automorphism group of an Okubo composition associated to a central simple algebra A is PGL_{1}(A). Therefore, these compositions are also called symmetric compositions of type ^{1}A_{2}.

Suppose next F is a field of characteristic different from 3 that does not contain a primitive cube root of unity. Let ω be a primitive cube root of unity in some separable closure of F, and let K = F (ω), a separable quadratic extension of F. Let B be a central simple K-algebra of degree 3 with a unitary involution τ leaving F fixed. We let Sym(B, τ)^{0} denote the F -vector space of τ-symmetric elements of reduced trace zero. Formula (9.3) defines a multiplication on Sym(B, τ)^{0}, and the form n of (9.4) is a quadratic form on Sym(B, τ)^{0}.

The following result is proved in [KMRT98, (34.35)]:

Proposition 9.6. The quadratic space (Sym(B, τ)^{0}, n) is a 3-fold Pfister quadratic space that becomes hyperbolic over K, and ⋆ is a normalized symmetric composition on this space.

Normalized symmetric compositions as in Proposition 9.6 are also called Okubo compositions. The automorphism group of an Okubo composition associated to a unitary involution τ on a central simple K-algebra B is PGU(B, τ). Therefore, these compositions are also called symmetric compositions of type ^{2}A_{2}.

Theorem 9.7. Let (S, n) be a 3-fold Pfister quadratic space over a field F of characteristic different from 3. Let ω be a primitive cube root of unity in a separable closure of F.

(1) If ω ∈ F, then every normalized symmetric composition of type II on (S, n) is isomorphic to the Okubo composition associated to some central simple F -algebra of degree 3, uniquely determined up to isomorphism. Such compositions exist if and only if n is hyperbolic.

(2) If ω ∉ F, let K = F (ω). Every normalized symmetric composition of type II on (S, n) is isomorphic to the Okubo composition associated to some central simple K-algebra of degree 3 with unitary involution, uniquely determined up to isomorphism. Such compositions exist if and only if n is split by K.

Proof. See Elduque-Myung [EM93, p. 2487] or [KMRT98, (34.37)]. □

Now, suppose the characteristic of F is 3. Normalized symmetric compositions of type II over F are extensively discussed in [EP96], [Eld97], [Eld99] and [Eld00]. They can be viewed as a specialization of symmetric compositions of type II over fields containing a primitive cube root of unity. To introduce them we first consider symbol algebras of degree 3. Any central simple algebra A of degree 3 over a field containing a cubic primitive root of unity ω is a symbol algebra, i.e., A has generators x, y such that x^{3} = a, y^{3} = b and yx = ω xy for some a, b ∈ F^{×}. We consider the Okubo composition ⋆ associated with A. The following basis of A^{0}:

(9.8)  \begin{matrix}e_{1} = x & e_{2} = y & e_{3} = ω^{2} xy & e_{4} = ω xy^{−1} \\ f_{1} = x^{−1} & f_{2} = y^{−1} & f_{3} = ω^{2} x^{−1}y^{−1} & f_{4} = ω x^{−1}y \end{matrix}

is hyperbolic for the form n and the multiplication table with respect to this basis of the product ⋆ is given by

e_{1}  f_{1}  e_{2}  f_{2}  e_{3}  f_{3}  e_{4}  f_{4}
e_{1} af_{1}  0  0  −e_{4}  0  −f_{2}  −af_{3}  0
f_{1} 0  a^{−1}e_{1} −f_{4}  0  −e_{2}  0  0  −a^{−1}e_{3}
e_{2} −e_{3}  0  bf_{2}  0  −be_{4}  0  −e_{1}  0
f_{2} 0  −f_{3}  0  b^{−1}e_{2}  0  −b^{−1}f_{4}  0  −f_{1}
e_{3} −af_{4}  0  0  −e_{1} abf_{3}  0  0  −bf_{2}
f_{3} 0 −a^{−1}e_{4} −f_{1}  0  0 a^{−1}b^{−1}e_{3} −b^{−1}e_{2}  0
e_{4} 0  −f_{2}  0 −b^{−1}e_{3} −af_{1}  0  a^{−1}bf_{4}  0
f_{4} −e_{2}  0  −bf_{3}  0  0  −a^{−1}e_{1}  0  ab^{−1}e_{4}

This multiplication table does not involve ω, hence specialising gives a symmetric composition ⋄_{a,b} over arbitrary fields, also of characteristic 3.

Proposition 9.9. 1) The symmetric composition ⋄_{1,1} is isomorphic to the split Petersson composition.

2) Let \overline{F} be an algebraic closure of F and let c, d ∈ \overline{F} be such that c^{3} = a and d^{3} = b. The symmetric compositions ⋄_{a,b} and ⋄_{1,1} are isomorphic over F (c, d). Thus the symmetric composition ⋄_{a,b} splits over F (c, d).

Proof. 1) and 2) follow by comparing multiplication tables. □

Theorem 9.10. Let (S, n) be a 3-fold Pfister quadratic space over a field F of characteristic 3. Every normalized symmetric composition of type II on (S, n) is isomorphic to a symmetric composition of the form ⋄_{a,b} for some a, b ∈ F^{×}.

Proof. The claim follows from [Eld99, p. 291] and a comparison of multiplication tables. □

Conditions for isomorphism ⋄_{a,b} ⟶^{∼} ⋄_{a′,b′} are discussed in [Eld99].

The automorphism groups (as group schemes) of the compositions ⋄_{a,b} are not smooth in characteristic 3 and are, as far as we know, not studied in the literature. Their groups of rational points are described in [Eld99].

Observe that symmetric compositions of type II in characteristic 3 are not necessarily split over separably closed fields, in contrast to symmetric compositions of type I and to symmetric compositions of type II in characteristic different from 3.

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Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton, AB T6G 2G1, Canada

   E-mail address: chernous@math.ualberta.ca

Department Mathematik, ETH Zentrum, CH-8092 Zürich, Switzerland

   E-mail address: knus@math.ethz.ch

ICTEAM Institute, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium, and Zukunftskolleg, Universität Konstanz, D-78457 Konstanz, Germany

   E-mail address: jean-pierre.tignol@uclouvain.be