THE KNESER-TITS CONJECTURE FOR GROUPS WITH TITS-INDEX E_{8,2}^{66} OVER AN ARBITRARY FIELD

R. PARIMALA, J.-P. TIGNOL, AND R. M. WEISS

Abstract. We prove: (1) The group of multipliers of similitudes of a 12dimensional anisotropic quadratic form over a field K with trivial discriminant and split Clifford invariant is generated by norms from quadratic extensions E/K such that q_{E} is hyperbolic. (2) If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is E_{8,2}^{66}, then G is generated by its root groups, as predicted by the Kneser-Tits conjecture.

1. Introduction

The Kneser-Tits conjecture—first formulated in [21]—predicts that the group of K-rational points (for some field K of arbitrary characteristic) of an absolutely simple algebraic group with Tits index

is generated by its root groups. This Tits-index is denoted by E_{8,2}^{66} in [19]. Groups with this Tits index are classified by similarity classes of anisotropic 12-dimensional quadratic forms over K with trivial discriminant and split Clifford invariant. By [22, 42.6], they are also the groups whose corresponding spherical building is a Moufang quadrangle of type E_{8} as defined in [22, 16.6].

Given a quadratic form q defined over a field K, we denote by clif(q) the Clifford invariant of q, by G(q) the group of multipliers of similitudes of q, by Hyp(q) the subgroup of K^{×} generated by K^{×2} and the norms from finite extensions E/K such that q_{E} is hyperbolic and by Hyp_{2}(q) the subgroup of Hyp(q) generated by K^{×2} and the norms from quadratic extensions E/K such that q_{E} is hyperbolic (including inseparable ones).

Our goal is to prove the following closely related statements.

Theorem 1.1. If q is an anisotropic quadratic form with trivial discriminant, then G(q) = Hyp_{2}(q) in the following cases:

(i) dim q = 8 and the index of clif(q) is 2; (ii) dim q = 12 and clif(q) is split.

Theorem 1.2. If G is the group of K-rational points of an absolutely simple algebraic group whose Tits index is E_{8,2}^{66}, then G is generated by its root groups.

Date: August 19, 2010.

2000 Mathematics Subject Classification. 11E04, 20G15, 20G41, 51E12.

Key words and phrases. Kneser-Tits conjecture, Moufang quadrangles, exceptional groups, multipliers of similitudes, R-equivalence.

We give two very different proofs of these theorems. In §2 we lay the groundwork that is common to the two proofs, and show that the equality G(q) = Hyp(q) holds for quadratic forms as in Theorem 1.1. As a consequence, the connected component of the identity PGO_{+}(q) in the group of projective similitudes is R-trivial if char(K) ≠ 2: see Corollary 2.19. In §3, we give proofs of Theorems 1.1 and 1.2 based on results in [23] and [24]. In particular, the notion of a quadrangular algebra introduced in Chapters 12–13 of [22] and in [23] plays a central role in these proofs. In §4 we show how the R-triviality of PGO_{+}(q) for q as in Theorem 1.1(ii) in characteristic 0 yields another proof of Theorem 1.2 in arbitrary characteristic. In §5 we give an entirely different proof of Theorem 1.1 under the assumption that char(K) ≠ 2, using the triality-defined correspondence between 8-dimensional quadratic forms of trivial discriminant and hermitian forms over the simple components of their even Clifford algebra.

For a survey of what is known about the Kneser-Tits conjecture; see [9]. We call attention especially to §6 of that paper, where the Kneser-Tits conjecture over an arbitrary field is discussed. By [17], [9, 6.1] and Theorem 1.2, the only exceptional groups of relative rank at least 2 for which the Kneser-Tits conjecture remains to be verified over arbitrary fields are those whose Tits index is

(called E_{8,2}^{78} in [19]). Groups with this Tits index are classified by isotopy classes of Albert division algebras, and the corresponding spherical buildings are the Moufang hexagons defined in [22, 16.8] for “hexagonal systems” of dimension 27. See also [9, 8.6] and [22, 37.41].

Acknowledgement. The proof in §4 that the R-triviality in characteristic 0 of PGO_{+}(q) for q as in Theorem 1.1(ii) implies Theorem 1.2 in arbitrary characteristic is due to Skip Garibaldi. We would like to thank him for allowing us to reproduce his proof here.

2. Similitudes of quadratic forms

Our main background reference for quadratic forms is [6], although we mostly use the notation of [23]. Let (K, L, q) be a quadratic space. Thus K is a field, L is a K-vector space and q: L → K is a quadratic form on L. We let f = ∂q denote the polar bilinear form of q. Thus

f(x, y) = q(x + y) − q(x) − q(y) for x, y ∈ L.

The quadratic space (K, L, q) is nondegenerate if dim_{K} rad f ≤ 1; see [6, 7.17]. If dim_{K} L is even and (K, L, q) is nondegenerate, then f is nondegenerate, the discriminant disc(q) is the isomorphism class of the center of the even Clifford algebra C_{0}(q) and the Clifford invariant clif(q) is the Brauer class of the full Clifford algebra C(q); see [6, §§13, 14]. As in [6, §§8, 9], we let I_{q}K denote the quadratic Witt group of K and let I_{q}^{n}K = I^{n−1}K · I_{q}K for all n > 0, where I^{n−1}K is the (n − 1)st power of the fundamental ideal IK of even-dimensional forms in the bilinear Witt ring W K.

The following definitions are taken from [22, 21.31].

THE KNESER-TITS CONJECTURE FOR E_{8,2}^{66} 3

Definition 2.1. A quadratic space (K, L, q) is of type E_{7} if it is anisotropic and there exists a separable quadratic extension E/K with norm N and scalars α_{1}, …, α_{4} such that

(K, L, q) ≅ (K, E^{4}, α_{1}N ⊥ α_{2}N ⊥ α_{3}N ⊥ α_{4}N)

and

α_{1}α_{2}α_{3}α_{4} ∉ N(E).

In other words, q is anisotropic,

q ≅ ⟨α_{1}, α_{2}, α_{3}, α_{4}⟩ · N,

and the quaternion algebra (E/K, α_{1}α_{2}α_{3}α_{4}), which represents clif(q), is not split.

Definition 2.2. A quadratic space (K, L, q) is of type E_{8} if it is anisotropic and there exists a separable quadratic extension E/K with norm N and scalars α_{1}, …, α_{6} such that

(K, L, q) ≅ (K, E^{6}, α_{1}N ⊥ α_{2}N ⊥ ⋯ ⊥ α_{6}N)

and

−α_{1}α_{2} ⋯ α_{6} ∈ N(E).

In other words, q is anisotropic,

q ≅ ⟨α_{1}, …, α_{6}⟩ · N,

and clif(q) is split.

Proposition 2.3. Suppose that (K, L, q) is an anisotropic quadratic space. Then the following hold:

(i) (K, L, q) is of type E_{7} if and only if dim q = 8, disc(q) is trivial and clif(q) is of index 2.

(ii) (K, L, q) is of type E_{8} if and only if dim q = 12, disc(q) is trivial and clif(q) is split. These conditions are also equivalent to dim q = 12 and q ∈ I_{q}^{3}K.

Proof. If char(K) ≠ 2, (i) is in [11, Ex. 9.12] and (ii) in [16, p. 123]. In arbitrary characteristic, see [5, 4.12] and (for the second part of (ii)) [6, Thm. 16.3] if char(K) = 2. □

Remark 2.4. Suppose that (K, L, q) is a quadratic space of type E_{7} and that q(1) = 1 for a distinguished element 1 of L. Let E and α_{1}, …, α_{4} be as in 2.1. By [23, 2.24],

(2.5)  C(q, 1) ≅ M(4, D) ⊕ M(4, D),

where C(q, 1) is the Clifford algebra with base point as defined in [22, 12.47] and D is the quaternion division algebra (E/K, α_{1}α_{2}α_{3}α_{4}). By [22, 12.51], C(q, 1) is isomorphic to the even Clifford algebra C_{0}(q). Since D represents clif(q), it is independent of the choice of the orthogonal decomposition of q in 2.1.

Our goal in this section is to prove the equality G(q) = Hyp(q) for q of type E_{7} or E_{8}. We start with some general observations. The following is essentially [23, 2.18].

Proposition 2.6. Suppose that the polynomial p(x) = x^{2} − αx + β ∈ K[x] is separable and irreducible over K. Let E be the splitting field of p(x) over K and let N denote the norm of the extension E/K, so that (K, E, N) is a nondegenerate anisotropic 2-dimensional quadratic space. Let (K, L, q) be a finite-dimensional quadratic space. Then the following assertions are equivalent:

(i) The K-vector space structure on L extends to an E-vector space structure such that q(u · v) = N(u)q(v) for all u ∈ E, v ∈ L;

(ii) There exists a similitude T of q such that q(T (v)) = βq(v) and f(v, T (v)) =

α for all non-zero v ∈ L and p(T) = 0; (iii) For each v_{1} ∈ L there exists a decomposition L = V_{1} ⊕ ⋯ ⊕ V_{d} for some

d ∈ ℕ such that v_{1} ∈ V_{1}, the restriction q_{i} of q to V_{i} is similar to N for each i ∈ [1, d] and q = q_{1} ⊥ ⋯ ⊥ q_{d};

(iv) for some d ∈ ℕ and some α_{1}, α_{2}, …, α_{d} ∈ K^{×},

(K, L, q) ≅ (K, E^{d}, α_{1}N ⊥ α_{2}N ⊥ ⋯ ⊥ α_{d}N);

(v) q_{E} is hyperbolic.

Proof. Suppose that (i) holds, choose a root γ ∈ E of p(x) and let T (v) = γ · v for all v ∈ L. Then T is a similitude of q as in (ii). If T is a similitude of q as in (ii), then for each nonzero v ∈ L the restriction of q to ⟨v, T (v)⟩ is similar to N (and, in particular, is nondegenerate). Therefore (iii) holds, and (iii) of course implies (iv). Fixing an isomorphism as in (iv), we may transfer to L the natural E-vector space structure on E^{d} to obtain (i). The equivalence of (iv) and (v) follows readily from [6, Prop. 34.8]. □

Definition 2.7. A similitude φ of a quadratic space (K, L, q) is called inseparable if char(K) = 2, the multiplier of φ is not in K^{×2} and

f(v, φ(v)) = 0 for all v ∈ L,

where f = ∂q. We call a similitude of q separable if it is not inseparable. Thus, if char(K) ≠ 2 all similitudes are separable.

Proposition 2.8. Let (K, L, q) be a finite-dimensional quadratic space such that f = ∂q is nondegenerate. If (K, L, q) admits an inseparable similitude with multiplier γ, then

q ≃ ⟨1, γ⟩ · q_{0}

for some non-degenerate quadratic form q_{0}. In particular, dim L ≡ 0 mod 4 and q_{K(\sqrt{γ})} is hyperbolic.

Proof. Let E = K(\sqrt{γ}) be a purely inseparable quadratic extension of K, and let φ be an inseparable similitude of (K, L, q) with multiplier γ. Linearizing the condition f(v, φ(v)) = 0, we obtain

f(v, φ(w)) = f(φ(v), w) for all v, w ∈ L.

Since f(φ(v), φ(w)) = γf(v, w) for all v, w ∈ L, it follows that

f(v, φ^{2}(w)) = f(φ(v), φ(w)) = γf(v, w) for all v, w ∈ L,

hence φ^{2}(w) = γw for all w ∈ L. We then define on L an E-vector space structure by

(λ + μ\sqrt{γ}) · v = λv + μφ(v) for λ, μ ∈ K and v ∈ L,

and we define a map f′: L × L → E by

f′(v, w) = f(v, w) + \sqrt{γ^{−1}}f(v, φ(w)) for v, w ∈ L.

A straightforward computation shows that f′ is a bilinear alternating form on L. It is nondegenerate since f is nondegenerate. Therefore, the dimension of L over E is even, hence its dimension over K is a multiple of 4. Let (e_{i}, e′_{i})_{i=1}^{d} be a symplectic E-base of L for f′. If L_{0} ⊂ L is the K-span of (e_{i}, e′_{i})_{i=1}^{d} and q_{0} is the restriction of q to L_{0}, we have L = L_{0} ⊥ φ(L_{0}) and q = q_{0} ⊥ ⟨γ⟩q_{0}. □

Corollary 2.9. Let (K, L, q) be a finite-dimensional quadratic space such that ∂q is non-degenerate. Then the multiplier of every inseparable similitude of q is in Hyp_{2}(q).

Proof. Let γ be the multiplier of an inseparable similitude of q. Clearly, γ ∈ N(K(\sqrt{γ})), and Proposition 2.8 shows that q is hyperbolic over K(\sqrt{γ}). □

We now consider quadratic forms of low dimension. The following result is presumably well-known:

Proposition 2.10. Every 10-dimensional quadratic form in I_{q}^{3}(K) is isotropic.

Proof. This was proved by Pfister [16, p. 123] under the hypothesis that char(K) ≠ 2. The arguments also apply when char(K) = 2; see [5, Thm. 4.10]. □

We now consider quadratic spaces of type E_{7}. For the next statement, we do not require the form to be anisotropic.

Lemma 2.11. Let (K, L, q) be a nondegenerate quadratic space of dimension 8. If disc(q) is trivial and clif(q) is represented by a quaternion algebra Q with norm form N_{Q}, then q is Witt-equivalent to the sum of a multiple of N_{Q} and a multiple of some 3-fold Pfister quadratic form π: there exist α, β ∈ K^{×} such that

(2.12)  q = ⟨α⟩ · N_{Q} + ⟨β⟩ · π  in I_{q}K.

Moreover, G(q) = G(N_{Q}) ∩ G(π).

Proof. Let α ∈ K^{×} be a value represented by q. Consider the form

q′ = q ⊥ ⟨−α⟩ · N_{Q}.

This 12-dimensional form is isotropic and has trivial discriminant and Clifford invariant, hence it is in I_{q}^{3}K and is Witt-equivalent to a 10-dimensional form. By Proposition 2.10, it is actually equivalent to an 8-dimensional form. By the Arason– Pfister Hauptsatz [6, Thm. 23.7], this 8-dimensional quadratic form becomes hyperbolic over the function field of the corresponding quadric, hence it is a multiple of some 3-fold Pfister quadratic form π by [6, Cor. 23.4]. Letting q′ = ⟨β⟩ · π in I_{q}K, we have (2.12).

Now, for γ ∈ G(q) we have ⟨1, −γ⟩ · q = 0 in I_{q}K, hence

⟨1, −γ⟩ · ⟨α⟩ · N_{Q} = −⟨1, −γ⟩ · ⟨β⟩ · π in I_{q}K.

Since the left side is a form of dimension 8 and the right side is a form of dimension 16, the right side must be isotropic. It is then hyperbolic by [6, Cor. 9.10], since it is a multiple of a Pfister form. The left side is then also hyperbolic, which means that γ is in G(π) and in G(N_{Q}). We have thus proved G(q) ⊂ G(N_{Q})∩G(π). Since the reverse inclusion is clear, the proof is complete. □

Lemmas 2.13 and 2.14 are well-known when char(K) ≠ 2; see [7, 2.13] for Lemma 2.13. The proofs we give below do not require any separability hypothesis.

Lemma 2.13. Let E_{1}, E_{2} be linearly disjoint quadratic extensions of a field K, and let M = E_{1} ⊗_{K} E_{2}. The norm groups of E_{1}, E_{2}, and M are related as follows:

N(E_{1}/K) ∩ N(E_{2}/K) = K^{×2} · N(M/K).

Proof. Since K^{×2} ⊂ N(E_{i}/K) and N(M/K) ⊂ N(E_{i}/K) for i = 1, 2, the inclusion N(E_{1}/K)∩N(E_{2}/K) ⊃ K^{×2} ·N(M/K) is clear, and it suffices to prove the reverse inclusion. We identify E_{1} and E_{2} with subfields of M and consider α ∈ N(E_{1}/K)∩ N(E_{2}/K). Let x_{1} ∈ E_{1}^{×}, x_{2} ∈ E_{2}^{×} be such that

α = N_{E_{1}/K}(x_{1}) = N_{E_{2}/K}(x_{2}).

Let T_{E_{i}/K}: E_{i} → K be the trace map, for i = 1, 2. Computation shows that

x_{1}N_{M/E_{1}}(1 + x_{1}^{−1}x_{2}) = T_{E_{1}/K}(x_{1}) + T_{E_{2}/K}(x_{2}).

If x_{1} ≠ −x_{2}, the left side is nonzero. Taking the norm from E_{1} to K of each side yields

αN_{M/K}(1 + x_{1}^{−1}x_{2}) = (T_{E_{1}/K}(x_{1}) + T_{E_{2}/K}(x_{2}))^{2} ∈ K^{×2},

hence α ∈ K^{×2} · N(M/K). If x_{1} = −x_{2}, then x_{1} ∈ E_{1} ∩ E_{2} = K, hence α ∈ K^{×2}. □

Lemma 2.14. Any multiplier of similitude of an anisotropic quadratic Pfister space (K, L, π) is a square in K or is the norm of a quadratic extension over which π is hyperbolic.

Proof. By [6, Cor. 9.9], the multipliers of π are the represented values of π, so any γ ∈ G(π) has the form γ = π(v) for some v ∈ L. Let e ∈ L be such that π(e) = 1. If e and v are not linearly independent, then γ ∈ K^{×2}. Otherwise, let V be the K-span of e and v. The restriction of π to V is the norm form of a quadratic extension of K over which π is isotropic, hence hyperbolic by [6, Cor. 9.10]. By construction, this norm form represents γ. □

Proposition 2.15. For any nondegenerate quadratic space (K, L, q) of dimension 8 such that disc(q) is trivial and clif(q) has index 1 or 2, we have G(q) = Hyp(q).

Proof. It suffices to show G(q) ⊂ Hyp(q), since the reverse inclusion follows from the similarity norm principle [6, Thm. 20.14]. Let γ ∈ G(q), and consider a decomposition of q as in (2.12). We may assume q is not hyperbolic, otherwise Hyp(q) = K^{×} = G(q) and there is nothing to prove. If N_{Q} or π is isotropic, hence hyperbolic, the proposition readily follows from Lemma 2.14. For the rest of the proof, we may thus assume N_{Q} and π are anisotropic. By Lemma 2.11 we have γ ∈ G(N_{Q})∩G(π), hence Lemma 2.14 yields quadratic extensions E_{1}, E_{2} of K that split N_{Q} and π respectively, such that γ ∈ N(E_{1}/K) ∩ N(E_{2}/K). If E_{1} ≅ E_{2}, then E_{1} splits N_{Q} and π, hence also q. Since γ ∈ N(E_{1}/K), it follows that γ ∈ Hyp(q). If E_{1} ≅ E̸_{2}, then E_{1} and E_{2} are linearly disjoint over K, and the tensor product M = E_{1} ⊗_{K} E_{2} is a field that splits N_{Q} and π, hence also q. Since γ is a norm from E_{1} and from E_{2}, Lemma 2.13 shows that γ ∈ K^{×2} · N(M/K), hence γ ∈ Hyp(q). □

Proposition 2.15 applies in particular to quadratic spaces of type E_{7}. We now turn to spaces of type E_{8}.

Proposition 2.16. Suppose that (K, L, q) is of type E_{8}, that γ is the multiplier of a separable similitude of q and that γ ∉ K^{×2}. Then there exists a decomposition

(K, L, q) = (K, L_{1}, q_{1}) ⊥ (K, L_{2}, q_{2})

such that q_{2} is of type E_{7}, q_{1} is similar to the reduced norm of the quaternion division algebra representing clif(q_{2}) and γ is a multiplier of similitudes of both q_{1} and q_{2}.

Proof. Let φ be a separable similitude of q with multiplier γ. If char(K) = 2, we choose v ∈ L such that f(v, φ(v)) ≠ 0; if char(K) ≠ 2, we let v be an arbitrary non-zero vector in L. Next we set W = ⟨v, φ(v)⟩. Since γ ∉ K^{×2}, we have dim_{K} W = 2. Let q̂_{1} denote the restriction of q to W and let q̂_{2} denote the restriction of q to W^{⊥}. The form q̂_{1} is similar to the norm N of a quadratic extension E/K such that γ ∈ N(E). Since q̂_{1} is nondegenerate, the extension E/K is separable and q = q̂_{1} ⊥ q̂_{2}. Since q_{E} and (q̂_{1})_{E} have trivial discriminant and split Clifford invariant, also (q̂_{2})_{E} has trivial discriminant and split Clifford invariant. By Proposition 2.10, it follows that (q̂_{2})_{E} is isotropic. Hence we can choose a 2-dimensional subspace U of W^{⊥} such that the restriction of q̂_{2} to U is hyperbolic over E. By Proposition 2.6, the restriction of q̂_{2} to U is similar to N. Let L_{1} = W ⊕ U, let L_{2} = L_{1}^{⊥} and let q_{i} denote the restriction of q to L_{i} for i = 1 and 2. Then q_{1} is similar to the reduced norm of a quaternion division algebra D, γ is a multiplier of q_{1} and

(K, L, q) = (K, L_{1}, q_{1}) ⊥ (K, L_{2}, q_{2}).

Since γ is a multiplier of both q and q_{1}, both ⟨1, −γ⟩·q and ⟨1, −γ⟩·q_{1} are hyperbolic. Since

⟨1, −γ⟩ · q = ⟨1, −γ⟩ · q_{1} ⊥ ⟨1, −γ⟩ · q_{2},

it follows by Witt’s Cancellation Theorem that the product ⟨1, −γ⟩ · q_{2} is also hyperbolic. Hence q_{2} = ⟨γ⟩ · q_{2} in I_{q}K. By [6, 8.17], therefore, there is a similitude of q_{2} with multiplier γ. Since disc(q) and disc(q_{1}) are both trivial, so is disc(q_{2}). Furthermore, clif(q_{2}) = clif(q_{1}) since clif(q) is split. Since the quaternion division algebra D represents clif(q_{1}), it also represents clif(q_{2}). We conclude, in particular, that q_{2} is of type E_{7}. □

Corollary 2.17. For any nondegenerate quadratic space (K, L, q) of dimension 12 such that disc(q) and clif(q) are trivial, we have G(q) = Hyp(q).

Proof. As in Proposition 2.15, it suffices to prove G(q) ⊂ Hyp(q). If q is isotropic, then Proposition 2.10 shows that q is Witt-equivalent to a multiple of a 3-fold Pfister form, hence the inclusion follows from Lemma 2.14. For the rest of the proof, we may thus assume q is anisotropic, i.e., q is of type E_{8}.

Let γ ∈ G(q). If γ is the multiplier of an inseparable similitude, then we have γ ∈ Hyp(q) by Corollary 2.9. If γ is the multiplier of a separable similitude, we fix a decomposition q = q_{1} ⊥ q_{2} as in Proposition 2.16, so γ ∈ G(q_{1}) ∩ G(q_{2}). By Proposition 2.15 we have G(q_{2}) = Hyp(q_{2}). Now, if E/K is a finite extension such that (q_{2})_{E} is hyperbolic, then E splits clif(q_{2}). Hence (q_{1})_{E} is hyperbolic, and therefore q_{E} is hyperbolic. This shows Hyp(q_{2}) ⊂ Hyp(q). Since γ ∈ Hyp(q_{2}), it follows that γ ∈ Hyp(q). □

Remark 2.18. Restricting to quadratic extensions that split q_{2} in the last part of the proof above, we see that Hyp_{2}(q_{2}) ⊂ Hyp_{2}(q). This observation will be used in §5.

When char(K) ≠ 2, Proposition 2.15 and Corollary 2.17 yield information on the connected component of the identity PGO_{+}(q) in the group of projective similitudes of (K, L, q), which is the group of algebra automorphisms of End(L) that commute with the adjoint involution of q. The property of R-triviality used in the following statement refers to Manin’s R-equivalence; see [14, §1] for details.

Corollary 2.19. Assume char(K) ≠ 2. For q a quadratic form of type E_{7} or E_{8} over K, the group PGO_{+}(q) is R-trivial.

Proof. By [14, Thm. 1], it suffices to prove that G(q_{E}) = Hyp(q_{E}) for every field E containing K. If q is of type E_{7}, this property readily follows from Proposition 2.15. If it is of type E_{8}, it follows from Corollary 2.17. □

Remark 2.20. Skip Garibaldi has observed that if q is of type E_{7}, then it follows from Lemma 2.11 and [8, Prop. 6.1] that the group PGO_{+}(q) is actually stably rational.

3. Quadrangular algebras and proofs of Theorems 1.1 and 1.2

Most of this section is devoted to results about quadrangular algebras. At the very end of this section, we use these results to prove Theorems 1.1 and 1.2.

The notion of a quadrangular algebra arose in the course of the classification of Moufang polygons; see, in particular, Chapters 12–13 and 27 in [22]. For the definition, see [23, 1.17].

Proposition 3.1. Let (K, L, q) be a quadratic space of type E_{7} or E_{8} as defined in 2.1 and 2.2 and suppose that q(1) = 1 for a distinguished element 1 of L. Then there exists a unique quadrangular algebra

Ξ = (K, L, q, 1, X, ·, h, θ)

as defined in [23, 1.17].

Proof. Existence holds by [23, Thm. 10.1] and uniqueness (up to equivalence as defined in [23, Thm. 1.22]) holds by [23, 6.42]. □

Notation 3.2. For the rest of this section, we let

Ξ = (K, L, q, 1, X, ·, h, θ)

be as in 3.1 and f = ∂q. By [23, Prop. 4.2], we can assume that Ξ is δ-standard for some δ ∈ L as defined in [23, 4.1]. (This allows us to use the identities in Chapter 4 of [23].) In addition, we let σ be as [23, 1.2], we let u^{−1} for all non-zero u ∈ L be as in [23, 1.3] and we let π be as in [23, 1.17(D1)].

Remark 3.3. Suppose that (K, L, q) is of type E_{7} and let C(q, 1) and D be as in 2.4. By (2.5) and [23, 1.17(A1)-(A3) and Prop. 2.22], there exists a unique map ∗ from D × X to X with respect to which X is a left vector space over D, ta = t ∗ a for all (a, t) ∈ X × K and w ∗ (a · v) = (w ∗ a) · v for all w ∈ D, a ∈ X and v ∈ L. This map is given explicitly in [24, 3.6].

Proposition 3.4. Suppose that (K, L, q) is of type E_{7}, let D and ∗ be as in 3.3, let φ_{1} be a similitude of q, let u = φ_{1}(1) and let

a ˆ· v = (a · v) · u^{−1}

for all (a, v) ∈ X × L, where u^{−1} is as in 3.2. Then there exists a similitude φ with the same multiplier as φ_{1} such that u = φ(1), an element ω ∈ K^{×} and a D-linear automorphism ψ of X such that the following hold:

(i) ψ(a · v) = ψ(a) ˆ· φ(v) for all a ∈ X and all v ∈ L.

(ii) φ(h(a, b)) = ωh(ψ(a), ψ(b)u) for all a, b ∈ X.

(iii) φ(θ(a, v)) ≡ ωθ(ψ(a), φ(v)) (mod ⟨φ(v)⟩) for all a ∈ X and all v ∈ L.

Proof. The map φ_{1} is an isomorphism of pointed quadratic spaces from (K, L, q, 1) to (K, L, q/q(u), u). It therefore induces an isomorphism of Clifford algebras with base point from C(q, 1) to C(q/q(u), u). Let Ξ̂, ĥ and θ̂ be as in [23, Prop. 8.1]; thus, Ξ̂ = (K, L, q/q(u), u, X, ˆ·, ĥ, θ̂) is the isotope of Ξ at u as defined in [23, 8.7]. By [23, 1.17(A1)–(A3) and Prop. 2.22] applied to both Ξ and to Ξ̂, X is a right C(q, 1)-module with respect to · and a right C(q/q(u), u)-module with respect to ˆ·. By [22, 12.55] (where the base point 1 is called ϵ), exactly one of the two direct summands in (2.5) acts nontrivially on X, and by [22, 12.54], there exists an isometry ρ of q fixing 1 that extends to an automorphism of C(q, 1) interchanging the two direct summands. Thus for j = 0 or 1, the composition φ_{1} ∘ ρ^{j} maps the direct summand A of C(q, 1) acting nontrivially on X to the direct summand A_{u} of C(q/q(u), u) acting nontrivially on X. Let φ = φ_{1} ∘ ρ^{j}. Choosing a basis for X as a left vector space over D, we can identify both A and A_{u} with End_{D}(X). It follows that there exists a D-linear automorphism ψ of X such that (i) holds. By [23, 1.25 and Prop. 6.38], there exists ω ∈ K^{×} such that also (ii) and (iii) hold. □

Notation 3.5. Let g and ϕ be the maps that appear in [23, 1.17(C3)-(C4)], let

(U_{+}, U_{1}, U_{2}, U_{3}, U_{4})

be the root group sequence and x_{4} the isomorphism from L to U_{4} obtained by applying the recipe in [22, 16.6] to Ξ, g and ϕ, let Γ be the corresponding Moufang quadrangle (see [22, 8.11]), let G^{†} be the subgroup of Aut(Γ) generated by the root groups of Γ, let H_{0} be the subgroup of Aut(Γ) defined in [24, 1.4] (or [23, 11.20]), let G = H_{0} · G^{†} and let H^{†} = H_{0} ∩ G^{†}. By [22, 35.11], the similarity class of (K, L, q) is an invariant of Γ.

Proposition 3.6. G/G^{†} ≅ H_{0}/H^{†} and if (K, L, q) is of type E_{8}, then G is the group of K-rational points of an absolutely simple algebraic group with Tits index E_{8,2}^{66}, and every such group arises in this way starting with some quadratic space of type E_{8} defined over K.

Proof. For the isomorphism G/G^{†} ≅ H_{0}/H^{†}, see the top of page 193 of [24] (where G is called G_{0}), The remaining assertions hold by [22, 42.6]. □

Proposition 3.7. Suppose that (K, L, q) is of type E_{7} and that φ_{1} is a similitude of q. Let H_{0} and x_{4} be as in 3.5. Then there exist an element h of H_{0} and a similitude φ of q with the same multiplier as φ_{1} such that

x_{4}(v)^{h} = x_{4}(φ(v))

for all v ∈ L.

Proof. Let φ and ψ be the maps obtained by applying Proposition 3.4 to φ_{1}. By Proposition 3.4 and [23, Prop. 12.5], the pair (φ, ψ) is contained in the structure group of Ξ (as defined in [23, 12.4]). The claim follows now by [23, Thm. 12.11] and the first few lines of its proof (as well as [23, 11.22]). □

From now on we identify K with its image under the map t ↦ t · 1 from K to L. Thus when we write π(a) + t for (a, t) ∈ X × K, for example, we mean π(a) + t · 1 (where π is as in 3.2).

Proposition 3.8. Let a be a non-zero element of X, let

p(x) = x^{2} − f(1, π(a))x + q(π(a)) ∈ K[x],

let E be the splitting field of p(x) over K and let N be the norm of the extension E/K. Let T (v) = θ(a, v) for all v ∈ L and let I be the identity automorphism of L. Then N(E^{×}) = K^{×2} · {q(π(a) + t) | t ∈ K}, q_{E} is hyperbolic and for each t ∈ K, T + tI is a similitude of q with multiplier q(π(a) + t).

Proof. By [23, 1.17(D2)], p(t) = q(π(a) − t) ≠ 0 for each t ∈ K. Thus p(x) is irreducible over K. It follows that N(E^{×}) = K^{×2} · {q(π(a) + t) | t ∈ K}. By [23, Props. 4.9(i) and 4.22], T +tI is a similitude of q with multiplier q(π(a)+t) for each t ∈ K and f(T (v), v) = f(π(a), 1)q(v) for each non-zero v ∈ L. By [23, Prop. 4.21], p(T) = 0. Thus if p(x) is separable, then q_{E} is hyperbolic by Propositions 2.6. If p(x) is inseparable, then f(π(a), 1) = 0, hence T is inseparable and again q_{E} is hyperbolic, this time by Proposition 2.8. □

Definition 3.9. For each non-zero a ∈ X, the map v ↦ θ(a, v) is a similitude of q by Proposition 3.8. We call an element a ∈ X separable if a ≠ 0 and the similitude v ↦ θ(a, v) of q is separable as defined in 2.7. We let X_{sep} denote the set of separable elements of X. Thus if char(K) ≠ 2, then X_{sep} = X\{0}, but if char(K) = 2, then by [23, Prop. 4.9(i)],

X_{sep} = {a ∈ X | f(π(a), 1) ≠ 0}.

If char(K) = 2, then by [22, 13.42–13.43], a ↦ f(π(a), 1) is a nondegenerate quadratic form on X. In particular, the set X_{sep} is non-empty also if char(K) = 2.

Proposition 3.10. Every inseparable similitude of q (as defined in 2.7) is the product of two separable similitudes.

Proof. Let φ be an inseparable similitude of q, so char(K) = 2. It suffices to show that

f(θ(a, φ(v)), v) ≠ 0

for some v ∈ L and some a ∈ X_{sep}, where X_{sep} is as in 3.9. Suppose this is false and let w = φ(1). Then

(3.11)  f(θ(a, w), 1) = 0

for all a ∈ X_{sep}. Furthermore,

(3.12)  f(w, 1) = 0

but w ∉ ⟨1⟩ since φ is inseparable. Choose a ∈ X_{sep}. Since f is nondegenerate, we can choose v ∈ ⟨w⟩^{⊥}\⟨1⟩^{⊥}. Replacing v by v + w if necessary, we can assume in addition (by [23, Prop. 4.9(i)] again) that

(3.13)  f(θ(a, w), v) ≠ 0.

By [23, Prop. 3.21], av ∈ X_{sep}, so f(θ(av, w), 1)) = 0 by (3.11). By [23, 1.17(C4)] and (3.12), it follows that

f(θ(a, w^{σ})^{σ}, 1)q(v) = f(w, v^{σ})f(θ(a, v)^{σ}, 1) + f(θ(a, v), w^{σ})f(v^{σ}, 1),

where σ is as in 3.2. By [23, 1.4] and (3.12), we have x^{σ} = x for x = 1 and x = w and f(x^{σ}, y) = f(x, y^{σ}) for all x, y ∈ L. Therefore

f(θ(a, w^{σ})^{σ}, 1) = f(θ(a, w), 1) = 0

by (3.11) and

f(w, v^{σ}) = f(w^{σ}, v) = f(w, v) = 0

by the choice of v and hence

f(θ(a, v), w)f(v, 1) = f(θ(a, v), w^{σ})f(v^{σ}, 1) = 0.

Since f(v, 1) ≠ 0 by the choice of v, we conclude that f(θ(a, v), w) = 0. By [23, Prop. 4.22], therefore, f(θ(a, θ(a, v)), θ(a, w)) = 0. By [23, Prop. 4.21], it follows that

f(π(a), 1)f(θ(a, v), θ(a, w)) = q(π(a))f(v, θ(a, w)).

By (3.13), therefore, f(θ(a, v), θ(a, w)) ≠ 0. By one more application of [23, Prop. 4.22], however, f(θ(a, v), θ(a, w)) = q(π(a))f(v, w) = 0. □

Proposition 3.14. Let E/K be a separable quadratic extension such that q_{E} is hyperbolic and let V_{i} and q_{i} for i ∈ [1, d] be as in Proposition 2.6(iii) with v_{1} = 1. Then there exists e ∈ X_{sep} such that θ(e, V_{i}) = V_{i} for each i ∈ [1, d].

Proof. Let p(x) = x^{2} − αx + β ∈ K[x] be an irreducible polynomial that splits over E. We can choose p(x) so that α = 0 if and only if char(K) ≠ 2. Let γ, γ_{1} ∈ E be the two roots of p(x). There exists an E-vector space structure on L as in Proposition 2.6(i) such that V_{i} is a 1-dimensional subspace for each i ∈ [1, d]. Let T (v) = γ · v for each v ∈ L and let T^{ϵ} be the unique automorphism of L such that T^{ϵ}(v) = γ_{1} · v for all v ∈ V_{1} and T^{ϵ}(v) = T (v) for all v ∈ V_{1}^{⊥}. Both T and T^{ϵ} are norm splitting maps of q as defined in [22, 12.14] and both map V_{i} to itself for each i ∈ [1, d]. By [22, 12.20 and 13.13(ii)], therefore, we can choose R ∈ {T, T^{ϵ}} such that R is linked to the map (a, v) ↦ a · v at some point e ∈ X as defined in [22, 13.2]. By [22, 13.61], e ∈ X_{sep} and there exists r ∈ K^{×} and s ∈ K such that R(v) = rθ(e, v) + sv for all v ∈ L. □

Proposition 3.15. Suppose that (K, L, q) is of type E_{8} and that

(K, L, q) = (K, L_{1}, q_{1}) ⊥ (K, L_{2}, q_{2})

with q_{2} of type E_{7}, q_{1} similar to the reduced norm of the quaternion division algebra representing clif(q_{2}) and 1 ∈ L_{2}. Then the following hold:

(i) There exists e ∈ X_{sep} such that θ(e, L_{i}) = L_{i} for i = 1 and 2.

(ii) Let e ∈ X be as in (i), let X_{e} be the subspace of X generated by elements of the form ev_{1}v_{2} ⋯ v_{j}, where v_{i} ∈ L_{2} for i ∈ [1, j] and j ≥ 1 is arbitrary, let ·_{e}, h_{e}, respectively, θ_{e} denote the restriction of ·, h, respectively, θ to X_{e} × L_{2}, X_{e} × X_{e}, respectively, X_{e} × L_{2} and let

Ξ_{e} = (K, L_{2}, q_{2}, 1, X_{e}, ·_{e}, h_{e}, θ_{e}).

Then Ξ_{e} is a quadrangular algebra.

Proof. By 2.2, 2.4 and Proposition 2.6, both (q_{1})_{E} and (q_{2})_{E} are hyperbolic. We can thus choose V_{i} and q_{i} for i ∈ [1, d] as in Proposition 2.6(iii) with v_{1} = 1, V_{i} ⊂ L_{2} for i ∈ [1, 4] and V_{i} ⊂ L_{1} for i ∈ [5, 6]. By Proposition 3.14, therefore, there exists e ∈ X_{sep} such that θ(e, L_{i}) = L_{i} for i = 1 and 2. Thus (i) holds.

Let Ξ_{e} be as described in (ii). To show that Ξ_{e} is a quadrangular algebra, it therefore suffices to show that X_{e} · L_{2} ⊂ X_{e}, θ(X_{e}, L_{2}) ⊂ L_{2} and h(X_{e}, X_{e}) ⊂ L_{2}. The first of these inclusions holds by the definition of X_{e}. To show the other two inclusions, we first choose non-zero elements v_{i} ∈ V_{i} for i ∈ [2, 5]. We can assume that e is the element of X chosen in [23, 6.4]. Thus the set 1, v_{2}, …, v_{5} is e-orthogonal as defined in [23, 6.6]. By [23, 1.17(A3) and Prop. 6.16], there exists a non-zero v_{6} ∈ L such that 1, v_{2}, …, v_{5}, v_{6} is e-orthogonal and ev_{2}v_{3}v_{4}v_{5}v_{6} = e. (We are not claiming that v_{6} ∈ V_{6} or even v_{6} ∈ L_{1}.) Let I_{2} be as in [23, 6.32], let J denote subset of I_{2} containing all the elements of I_{2} that are subsets of {v_{2}, v_{3}, v_{4}} together with the element {v_{5}, v_{6}} ∈ I_{2} (so |J| = 8), let J_{2} be the elements of J of cardinality 2 (so |J_{2}| = 4), let X_{x} for each x ∈ J be as in [23, 6.35], let M be the subspace of X spanned by {X_{m} | m ∈ J} and let N be the subspace of M spanned by {X_{m} | m ∈ J_{2}}. By [23, Prop. 6.34], dim_{K} M = 16 and M = eL_{2} ⊕ N. By [23, 6.37], we have M = X_{e}, by [23, Prop. 6.13], we have h(e, N) = 0 and by [23, Props. 3.15 and 4.5(i)], h(e, eL_{2}) ⊂ L_{2} (since θ(e, L_{2}) ⊂ L_{2}). Hence h(e, X_{e}) ⊂ L_{2}. By repeated application of [23, 1.17(B1)–(B2)], it follows that h(X_{e}, X_{e}) ⊂ L_{2}. Since 1 ∈ L_{2}, we have L_{2}^{σ} ⊂ L_{2}, where σ is as in 3.2. By repeated application of [23, 1.17(C3)–(C4)], it follows from θ(e, L_{2}) ⊂ L_{2} first that θ(eL_{2}, L_{2}) ⊂ L_{2} and then that θ(X_{e}, L_{2}) ⊂ L_{2}. Thus (ii) holds. □

Definition 3.16. For each non-zero u in L, let π_{u} be the reflection of q given by

π_{u}(v) = f(u, v)u/q(u) − v

for all v ∈ L. Thus π_{1} = σ, where σ is as in 3.2.

Proposition 3.17. Let H^{†} and x_{4} be as in 3.5. Suppose that φ is a product of an even number of reflections of q as defined in 3.16. Then there exists an element h ∈ H^{†} such that

x_{4}(v)^{h} = x_{4}(φ(v))

for all v.

Proof. Let u be a non-zero element of L. By [24, eq. (6)–(14)], there are elements w_{1}(0, q(u)) and w_{4}(u) in H^{†} such that

x_{4}(v)^{w_{1}(0,q(u))w_{4}(u)} = x_{4}(v/q(u))^{w_{4}(u)} = x_{4}(π_{u}π_{1}(v))

for each v ∈ L. □

Notation 3.18. Let M denote the subgroup of K^{×} generated by the non-zero elements in the set {q(π(a) + t) | (a, t) ∈ X × K}.

Thus

(3.19)  M ⊂ G(q) ∩ Hyp_{2}(q)

by Proposition 3.8 and K^{×2} = {q(π(a) + t) | (a, t) ∈ {0} × K^{×}} ⊂ M.

Proposition 3.20. Let H^{†} and x_{4} be as in 3.5. For each h ∈ H^{†}, there exists a unique similitude φ_{h} of q such that

x_{4}(v)^{h} = x_{4}(φ_{h}(v))

for all v ∈ L. Furthermore, the map h ↦ γ_{h} is a surjective homomorphism from H^{†} to M, where γ_{h} is the multiplier of φ_{h}.

Proof. Let w_{1}(a, t) for non-zero (a, t) ∈ X × K and w_{4}(u) for non-zero u ∈ L be as in [24, eqs. (6)–(7)]. Then

x_{4}(v)^{w_{4}(u)} = x_{4}(uf(u, v^{σ}) − q(u)v^{σ})

and

x_{4}(v)^{w_{4}(a,t)} = x_{4}((θ(a, v) + tv)/q(π(a) + t))

for all v ∈ L and all non-zero (a, t) ∈ X × K by [24, eqs. (13)–(14)]. We have

q((θ(a, v) + tv)/q(π(a) + t)) = q(v)/q(π(a) + t)

for all v ∈ L and all non-zero (a, t) ∈ X × K (by 3.8) and

q(uf(u, v^{σ}) − q(u)v^{σ}) = q(v)q(u)^{2}

for all u, v ∈ L since q(v^{σ}) = q(v). The claim holds, therefore, by [24, Thm. 2.1]. □

Proposition 3.21. If (K, L, q) is of type E_{7}, then G(q) = M.

Proof. By (3.19), it suffices to show that G(q) ⊂ M. Let φ_{1} be a similitude of q, let x_{4}, U_{4}, H_{0} and H^{†} ⊂ H_{0} be as in 3.5 and let h and φ be as in Proposition 3.7. Thus

x_{4}(v)^{h} = x_{4}(φ(v))

for each v ∈ L and φ is a similitude of q with the same multiplier as φ_{1}. Let H_{1} and H_{2} be the subgroups of H_{0} defined in [24, 3.12 and 3.14]. By [24, Thm. 3.15(ii)], H_{0} = H_{1}H_{2} and by [24, Thm. 5.19], H_{2} ⊂ H_{1}H^{†}. We conclude that H_{0} = H_{1}H^{†}. By [24, Prop. 3.11], H_{1} centralizes U_{4}. There thus exists g ∈ H^{†} such x_{4}(v)^{g} = x_{4}(φ(v)) for each v ∈ L. The claim holds, therefore, by Proposition 3.20. □

Proposition 3.22. If (K, L, q) is of type E_{8}, then G(q) = M.

Proof. By (3.19), it suffices to show that G(q) ⊂ M. Let φ be a similitude of q whose multiplier is not in K^{×2}. By Proposition 3.10, it suffices to assume that φ is separable. Let

(K, L, q) = (K, L_{1}, q_{1}) ⊥ (K, L_{2}, q_{2})

be the decomposition of q obtained by applying Proposition 2.16 to φ. Replacing Ξ by an isotope as defined in [23, 8.7], we can assume that the base point 1 lies in L_{2} (without changing the subgroup generated by the set of non-zero elements in {q(π(a) + t) | (a, t) ∈ X × K}). We can thus let e and

Ξ_{e} = (K, L_{2}, q_{2}, 1, X_{e}, ·_{e}, h_{e}, θ_{e})

with X_{e} ⊂ X be as in Proposition 3.15. By Proposition 3.21 (and the uniqueness assertion in Proposition 3.1), we conclude that γ is the product of elements in {q(π(a) + t) | (a, t) ∈ X_{e} × K}. □

We can now prove Theorems 1.1 and 1.2. By Proposition 2.3, a quadratic form satisfying the hypotheses of Theorem 1.1 is of type E_{7} or E_{8}. By the existence assertion in Proposition 3.1, we can apply all the results in this section. Hence G(q) ⊂ Hyp_{2}(q) by (3.19) and Propositions 3.21 and 3.22. By [6, Thm. 20.14], we have Hyp_{2}(q) ⊂ G(q). This concludes the proof of Theorem 1.1.

Suppose that (K, L, q) is of type E_{8} and that x_{4}, H_{0} and H^{†} are as in 3.5. To prove Theorem 1.2, it suffices by Proposition 3.6 (and the existence assertion in Proposition 3.1) to show that every element in H_{0} lies in H^{†}. Let h ∈ H_{0}. By [24, eq. (19)], there is a similitude φ of q such that

x_{4}(v)^{h} = x_{4}(φ(v))

for all v ∈ L. Replacing h by a suitable element in hH^{†}, we can assume, by Propositions 3.20 and 3.22, that φ is an isometry of q and hence a product of reflections of q. Again replacing h by a suitable element of hH^{†}, we can assume, by Proposition 3.17 and [24, Prop. 3.16], that φ is the identity. By [24, Thm. 3.12], h = α_{u} for some u ∈ C^{×}, where C = K by [24, 3.6] and α_{u} is as defined in [24, Prop. 3.11]. By [24, Prop. 3.13], it follows that h ∈ H^{†}. This concludes the proof of Theorem 1.2.

4. R-equivalence and an alternative proof of Theorem 1.2

In this section, we give an alternative proof of Theorem 1.2 based on Corollary 2.19 and various other results about R-equivalence. This proof is due to Skip Garibaldi. The methods employed in this section are completely different from those employed in the previous section; in particular, we make no further reference to the Moufang quadrangle Γ of §3.

Let G denote a reductive algebraic group of absolute type E_{8} whose Tits index over a field K is E_{8,2}^{66}. Our goal is to show that the group of K-rational points of G is generated by its root groups. By [9, 7.3], it suffices to assume that char(K) = 0. This will allow us to apply Corollary 2.19. By [9, 7.2], it suffices to show that G is R-trivial.

Now fix a maximal K-torus T containing a maximal K-split torus S in G and fix a pinning for G with respect to T over an algebraic closure of K. Number the simple roots α_{j} as in [3, Chapter 6, Plate VII] and let ω_{i}^{∨} be the corresponding fundamental dominant co-weights, so ⟨α_{j}, ω_{i}^{∨}⟩ = δ_{ij}. The fundamental co-weights ω_{1}^{∨} and ω_{8}^{∨} belong to the co-root lattice and so define cocharacters, in other words, homomorphisms from G_{m} to T. Their images generate a subtorus S in T which is the connected component of the intersection (in T) of the kernels of the roots α_{2}, …, α_{7}. There is a canonical isomorphism

(4.1)  Φ : K̄^{×} ⊗_{ℤ} T_{∗} ⟶^{∼} T (K̄),

where T_{∗} is the lattice of cocharacters of T and where K̄ is an algebraic closure of K; see [18, 3.2.11]. Since the group G is of adjoint type, the ω_{i}^{∨} form a ℤ-basis for T_{∗}. Thus we may view Φ as an isomorphism

∏_{i=1}^{8} K̄^{×} ⊗_{ℤ} ℤω_{i}^{∨} ⟶^{∼} T (K̄).

This shows that the intersection of the kernels of the roots α_{2} …, α_{7} is connected and that Φ restricts to an isomorphism

(4.2)  (K̄^{×} ⊗_{ℤ} ℤω_{1}^{∨}) × (K̄^{×} ⊗_{ℤ} ℤω_{8}^{∨}) ⟶^{∼} S(K̄).

The cocharacters ω_{1}^{∨} and ω_{8}^{∨} are defined over K by [2, Cor. 6.9], so (4.2) implies that S is K-isomorphic to the direct product of the images of the cocharacters ω_{1}^{∨} and ω_{8}^{∨}.

We next fix a parabolic P of G whose Levi subgroup is the connected reductive group Z_{G}(S); see [18, §13.4 and Lemma 15.1.2]. Let U be the unipotent radical of P and let U^{−} be the unipotent radical of the opposite parabolic. The product U^{−} × U is isomorphic as a variety to an affine space. By [1, Proof of Thm. 21.20], the natural map from G to G/P restricts to an isomorphism from U^{−} to an open subset of G/P. Hence the product map from U^{−} × P to G defines an isomorphism from U^{−} × P to an open subset of G. It follows that G is birationally equivalent to

U^{−} × Z_{G}(S) × U.

(This subvariety is the analog of the big cell for the Bruhat decomposition of G over K; see [2, Prop. 4.10(d)].) We conclude that G is birationally equivalent to the product of Z_{G}(S) and an affine space.

Let H denote the derived subgroup of Z_{G}(S). The sequence

1 → S → Z_{G}(S) → H/(H ∩ S) → 1

is exact on L-points for every extension L/K because S is split. Hence Z_{G}(S) is birationally equivalent to the product of S with H/(H ∩ S).

The absolute Dynkin diagram of H is of type D_{6}. By [20, p. 211], the group H is Spin(q) for q a quadratic form over K with dim q = 12, disc q = 1 and clif(q) split. As S centralizes H, the intersection H ∩ S is contained in the center μ_{2} × μ_{2} of Spin(q). We show that H ∩ S is equal to the center of Spin(q).

Since G is simply connected as well as adjoint, the co-roots α_{j}^{∨} provide also a ℤ-basis for the cocharacter lattice T_{∗}. Thus we may view the isomorphism Φ in (4.1) as an isomorphism

K̄^{×} ⊗_{ℤ} T_{∗} = ∏_{i=1}^{8} K̄^{×} ⊗_{ℤ} ℤα_{i}^{∨} ⟶^{∼} T (K̄).

Then it follows from [18, 8.1.8] that Φ restricts to an isomorphism

(4.3)  ∏_{i=2}^{7} K̄^{×} ⊗_{ℤ} ℤα_{i}^{∨} ⟶^{∼} (H ∩ T)^{0}(K̄).

The expressions for the fundamental dominant weights ω_{i} in terms of the roots α_{j} in [3, Chapter 6, Plate VII] imply expressions for the fundamental dominant co-weights ω_{i}^{∨} in terms of the co-roots α_{j}^{∨}. These expressions yield

ω_{1}^{∨}(−1) = α_{2}^{∨}(−1)α_{3}^{∨}(−1) and ω_{8}^{∨}(−1) = α_{3}^{∨}(−1)α_{5}^{∨}(−1)α_{7}^{∨}(−1).

From (4.2) and (4.3), we see that these two elements both lie in S(K̄) and in (H ∩ T)^{∘}(K̄) and are nontrivial and distinct. We have thus produced two distinct nontrivial elements in (H ∩ S)(K̄). Hence H ∩ S is, in fact, the entire center of Spin(q).

Therefore H/(H ∩S) is PGO_{+}(q). It follows by 2.19 that H/(H ∩S) is R-trivial. Therefore G is birationally equivalent to the product of PGO_{+}(q) times an affine space. Thus G itself is R-trivial by [4, p. 197, Cor.]. This concludes our second proof of Theorem 1.2.

We observe that this proof goes through verbatim for every group G of absolute type E_{8} in whose Tits index the roots α_{1} and α_{8} are circled. We conclude that for such groups, G is R-trivial and the group of K-rational points of G is generated by its root groups. With only minor modifications, the proof also shows that if G is adjoint of absolute type E_{7} with trivial Tits algebras and the root α_{1} is circled in the Tits index of G, then G is R-trivial.

5. Theorem 1.1 and triality

In this section, we assume that char(K) ≠ 2. We give an alternative proof of Theorem 1.1, based on completely different methods. We actually show:

Proposition 5.1. Suppose the characteristic of the base field K is different from 2. If q is a quadratic form with trivial discriminant, then Hyp_{2}(q) = Hyp(q) in the following cases:

(i) dim q = 8 and the index of clif(q) is 1 or 2; (ii) dim q = 12 and clif(q) is split.

Since in each case G(q) = Hyp(q) by Proposition 2.15 and Corollary 2.17, Theorem 1.1 follows from Proposition 5.1.

We start with the case of 8-dimensional quadratic forms. If clif(q) is split, then q is a multiple of a 3-fold Pfister form, and the result follows from Lemma 2.14. Similarly, if q is isotropic, then q is Witt-equivalent to a multiple of a 2-fold Pfister form, and the result follows from Lemma 2.14. We may thus assume that (K, L, q) is of type E_{7} and let D be the quaternion division algebra over K that represents clif(q). We show next that the Clifford algebra construction associates to q a skew-hermitian form h of rank 4 over D, and we shall complete the proof of Proposition 5.1(i) by proving that

Hyp(q) = Sn(h) = Hyp_{2}(q);

see Proposition 5.9.

Let (A, σ) be a central simple K-algebra of degree 8 with an orthogonal involution of trivial discriminant. The Clifford algebra C(A, σ) decomposes into a direct product of two central simple K-algebras of degree 8:

C(A, σ) = C_{+}(A, σ) × C_{−}(A, σ).

Recall that C(A, σ) carries a canonical involution \underline{σ}, which induces orthogonal involutions σ_{+} and σ_{−} on C_{+}(A, σ) and C_{−}(A, σ) respectively. By triality (see [12, (42.3)]), the Clifford algebras of (C_{+}(A, σ), σ_{+}) and (C_{−}(A, σ), σ_{−}) satisfy

(C(C_{+}(A, σ), σ_{+}), \underline{σ_{+}}) = (C_{−}(A, σ), σ_{−}) × (A, σ),
(C(C_{−}(A, σ), σ_{−}), \underline{σ_{−}}) = (A, σ) × (C_{+}(A, σ), σ_{+}).

Proposition 5.2. The following hold:

(1) If A is split, then (C_{+}(A, σ), σ_{+}) and (C_{−}(A, σ), σ_{−}) are isomorphic.

(2) If (A, σ) is split and isotropic, then (C_{+}(A, σ), σ_{+}) and (C_{−}(A, σ), σ_{−}) are hyperbolic.

(3) If (A, σ) is split and hyperbolic, then (C_{+}(A, σ), σ_{+}) and (C_{−}(A, σ), σ_{−}) are split and hyperbolic.

Proof. (1) is well-known, (2) is in [12, (8.5)], and (3) follows from (2) and the fact that the Clifford invariant of a hyperbolic quadratic form is trivial. □

We apply this proposition in the following context: let (K, L, q) be an 8-dimen-sional quadratic space with disc q = 1, and assume clif(q) is represented by a quaternion division algebra D. Let ad_{q}: End_{K} L → End_{K} L be the adjoint involution of q. We apply the discussion above with (A, σ) = (End_{K} L, ad_{q}). Then C(A, σ) = C_{0}(L, q) and (C_{+}(A, σ), σ_{+}), (C_{−}(A, σ), σ_{−}) are isomorphic to (End_{D} W, ad_{h}) for some 4-dimensional skew-hermitian space (W, h) over D (with its conjugation involution).

Proposition 5.3. For an arbitrary extension E/K, the following statements are equivalent:

(a) q_{E} is hyperbolic;

(b) D_{E} is split and (End_{D} W, ad_{h})_{E} is hyperbolic;

(c) D_{E} is split and (End_{D} W, ad_{h})_{E} is isotropic.

Proof. (a) ⇒ (b): This readily follows from Proposition 5.2(3).

(b) ⇒ (c): Clear.

(c) ⇒ (a): This follows from Proposition 5.2(2) with (End_{D} W, ad_{h})_{E} for (A, σ); then by triality (C_{+}(A, σ), σ_{+}) or (C_{−}(A, σ), σ_{−}) is isomorphic to (End_{K} L, ad_{q})_{E}). □

The next results 5.4–5.6 hold for skew-hermitian forms of arbitrary dimension.

Lemma 5.4. Let (W, h) be a skew-hermitian space over a quaternion division algebra D over K and let E be a quadratic extension of K. If h is anisotropic, the following conditions are equivalent:

(i) E ≅ K(h(v, v)) for some v ∈ W; (ii) D_{E} is split and h_{E} is isotropic.

Proof. If (i) holds, then E is isomorphic to a maximal subfield of D, hence D_{E} is split. Let h(v, v)^{2} = a ∈ K^{×}, so E ≅ K(\sqrt{a}). Then v · (h(v, v) + \sqrt{a}) ∈ W_{E} is isotropic for h_{E}. Thus, (ii) holds.

Conversely, if (ii) holds, then E is isomorphic to a maximal subfield of D. Let E = K(\sqrt{a}) for some a ∈ K, and let λ ∈ D be a pure quaternion such that λ^{2} = a. Suppose x + y\sqrt{a} ∈ W_{E} is h_{E}-isotropic for some x, y ∈ W. The condition h_{E}(x + y\sqrt{a}, x + y\sqrt{a}) = 0 yields

h(x, x) + h(y, y)a = 0 and h(x, y) + h(y, x) = 0.

Since h is skew-hermitian, the second equation shows that h(x, y) ∈ K. Then

h(x + yλ, x + yλ) = 2h(x, y)λ − h(y, y)a − λh(y, y)λ

and the right side commutes with λ. Therefore, h(x + yλ, x + yλ) = λb for some b ∈ K^{×}, and we have E ≅ K(h(v, v)) with v = x + yλ. □

For any skew-hermitian space (W, h) over a quaternion division algebra D over K, we let Sn(h) denote the group of spinor norms of h, which is the image of the Clifford group Γ(End_{D} W, ad_{h}) = Γ(W, h) under the multiplier map; see [12, (13.30)].

Proposition 5.5. If h is anisotropic, then Sn(h) = ∏_{E} N(E/K), where E runs over the quadratic extensions of K satisfying the equivalent conditions (i) and (ii) of Lemma 5.4.

Proof. The multiplier map Γ(W, h) → K^{×} factors through the vector representation Γ(W, h) → O_{+}(W, h), where O_{+}(W, h) is the group of direct isometries of the space (W, h). By [10, Thm. 6.2.17], this group is generated by transformations of the form

τ_{v,r}: W → W, x ↦ x − vh(vr, x)

where v ∈ W is an anisotropic vector and r ∈ D^{×} satisfies r − \overline{r} = rh(v, v)\overline{r}. To compute the spinor norm of that transformation, observe that τ_{v,r} is the identity on v^{⊥}, hence the spinor norm of τ_{v,r} is the spinor norm of its restriction to the 1-dimensional subspace vD. Let ν = h(v, v) ∈ D^{×} and let h_{v} denote the restriction of h to vD, so

h_{v}(vλ, vμ) = \overline{λ}νμ for λ, μ ∈ D.

We have

O_{+}(vD, h_{v}) = \{θ ∈ K(ν)^{×} | θ\overline{θ} = 1\} and Γ(vD, h_{v}) = K(ν)^{×}

(where θ ∈ K(ν)^{×} is identified with the map vλ ↦ vθλ for λ ∈ D). The vector representation Γ(vD, h_{v}) → O_{+}(vD, h_{v}) carries u ∈ K(ν)^{×} to u\overline{u}^{−1}, hence the spinor norm of that isometry is u\overline{u}K^{×2}; see [12, (13.17)]. This shows that Sn(h_{v}) consists of norms from the quadratic extension K(ν)/K. Since Sn(h) is generated by the groups Sn(h_{v}) for the anisotropic vectors v ∈ W, the proposition follows. □

Corollary 5.6. Let (W, h) be a skew-hermitian space over a quaternion division algebra D over K, and let p = char(K) > 2. For K̃ = K^{−p^{−∞}} the perfect closure of K, we have Sn(hK̃) ∩ K = Sn(h).

Proof. The inclusion Sn(h) ⊂ Sn(hK̃)∩K is clear, so it suffices to prove the reverse inclusion. Let x ∈ Sn(hK̃) ∩ K. If x ∈ K̃^{×2}, then x ∈ K^{×2} ⊂ Sn(h). We may thus assume x ∉ K̃^{×2}. By Proposition 5.5, there exist quadratic extensions Ẽ_{1}/K̃, …, Ẽ_{r}/K̃ such that DẼ_{i} is split and hẼ_{i} is isotropic for each i ∈ [1, r], and elements y_{i} ∈ Ẽ_{i} \ K̃ for i ∈ [1, r] such that

(5.7)  x = N_{Ẽ_{1}/K̃}(y_{1}) · . . . · N_{Ẽ_{r}/K̃}(y_{r}).

Let K′ ⊂ K̃ be the subfield generated by N_{Ẽ_{1}/K̃}(y_{1}), …, N_{Ẽ_{r}/K̃}(y_{r}) and, for i ∈ [1, r], let E′_{i} = K′(y_{i}). Thus, K′/K is a purely inseparable extension of finite degree, (5.7) yields

(5.8)  x = N_{E′_{1}/K′}(y_{1}) · . . . · N_{E′_{r}/K′}(y_{r}),

and each E′_{i}/K′ is a quadratic extension. For i ∈ [1, r], let E′′_{i} be the separable closure of K in E′_{i}; it is a quadratic extension of K and we have

E′_{i} ≅ E′′_{i} ⊗_{K} K′  and  Ẽ_{i} ≅ E′′_{i} ⊗_{K} K̃.

Since quaternion division algebras do not split over extensions of odd degree, the condition that DẼ_{i} is split shows that D_{E′′_{i}} is split. Likewise, anisotropic skew-hermitian forms do not become isotropic over odd-degree extensions by [15, Thm. 3.5], hence h_{E′′_{i}} is isotropic. Now, let [K′ : K] = p^{d}; taking the norm from K′ to K of each side of (5.8), we obtain

x^{p^{d}} = N_{E′_{1}/K}(y_{1})·. . .·N_{E′_{r}/K}(y_{r}) = N_{E′′_{1}/K}(N_{E′_{1}/E′′_{1}}(y_{1}))·. . .·N_{E′′_{r}/K}(N_{E′_{r}/E′′_{r}}(y_{r})).

Since x^{p^{d}} ≡ x mod K^{×2}, this equation shows that x is a product of norms from quadratic extensions over which D is split and h is isotropic, hence x ∈ Sn(h) by Proposition 5.5. □

We now return to the context of Proposition 5.3. The following proposition completes the proof of Proposition 5.1(i):

Proposition 5.9. For q an anisotropic 8-dimensional quadratic form and h the corresponding 4-dimensional skew-hermitian form as in Proposition 5.3, we have

Hyp(q) = Sn(h) = Hyp_{2}(q).

Proof. Proposition 5.3 shows that the quadratic extensions E/K such that D_{E} is split and h_{E} is isotropic are exactly those such that q_{E} is hyperbolic, hence by Proposition 5.5 we have

Sn(h) = Hyp_{2}(q) ⊂ Hyp(q).

To complete the proof, we show Hyp(q) ⊂ Sn(h). Let E/K be a finite-degree extension such that q_{E} is hyperbolic, let K̃ be the perfect closure of K in some algebraic closure of E, and let K_{1} be the purely inseparable closure of K in E. The compositum E · K̃ of E and K̃ satisfies E · K̃ ≅ E ⊗_{K_{1}} K̃. Since q_{E} is hyperbolic, q is also hyperbolic over E · K̃, hence D_{E·K̃} is split and h_{E·K̃} is isotropic, by Proposition 5.3. Therefore, Sn(h_{E·K̃}) = (E · K̃)^{×}. Since K̃ is perfect, we may apply the norm principle for spinor norms (see [13, (6.2)]), which is a twisted analogue of Knebusch’s norm theorem, to see that N(E · K̃/K̃) ⊂ Sn(hK̃). Since N(E/K_{1}) ⊂ N(E · K̃/K̃), it follows that N(E/K_{1}) ⊂ Sn(hK̃). Let p = char(K) if char(K) > 2 and p = 1 if char(K) = 0, so [K_{1} : K] = p^{d} for some d ≥ 0. For all x ∈ K_{1}, we have N_{K_{1}/K}(x) = x^{p^{d}}, hence

N(E/K) = N_{K_{1}/K}(N(E/K_{1})) = N(E/K_{1})^{p^{d}} ⊂ Sn(hK̃).

But N(E/K) ⊂ K, hence Corollary 5.6 shows that N(E/K) ⊂ Sn(h). Of course, we also have K^{×2} ⊂ Sn(h), hence Hyp(q) ⊂ Sn(h). □

Part (ii) of Proposition 5.1 follows from part (i) by the same arguments as in the proof of Corollary 2.17: let q be a 12-dimensional nondegenerate form with trivial discriminant and Clifford invariant. If q is isotropic, then it is Witt-equivalent to a 3-fold Pfister form and G(q) = Hyp_{2}(q) by Lemma 2.14. For the rest of the proof, suppose q is anisotropic, i.e., q is of type E_{8}. Let γ ∈ G(q). Since char(K) ≠ 2, all similitudes of q are separable. We can thus fix a decomposition q = q_{1} ⊥ q_{2} as in Proposition 2.16. Since γ ∈ G(q_{2}), part (i) of Proposition 5.1 shows that γ ∈ Hyp_{2}(q_{2}). Since Hyp_{2}(q_{2}) ⊂ Hyp_{2}(q) by Remark 2.18, it follows that γ ∈ Hyp_{2}(q). This proves Proposition 5.1(ii).

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Department of Mathematics and Computer Science, Emory University, MSC W401 400 Dowman Drive, Atlanta, GA 30322, USA

   E-mail address: parimala@mathcs.emory.edu

Institute for Information and Communication Technologies, Electronics and Applied Mathematics, Université Catholique de Louvain, Avenue Georges Lemaître, 4, 1348 Louvain-la-Neuve, Belgium

   E-mail address: Jean-Pierre.Tignol@uclouvain.be

Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, MA 02155, USA

   E-mail address: rweiss@tufts.edu