Let G be an algebraic group over a field k. We call g ∈ G(k) real if g is conjugate to g −1 in G(k). In this paper we study reality for groups of type G 2 over fields of characteristic different from 2. Let G be such a group over k. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element in G(k) is real if and only if it is a product of two involutions in G(k). Every unipotent element in G(k) is a product of two involutions in G(k). We discuss reality for G 2 over special fields and construct examples to show that reality fails for semisimple elements in G 2 over Q and Q p . We show that semisimple elements are real for G 2 over k with cd(k) ≤ 1. We conclude with examples of nonreal elements in G 2 over k finite, with characteristic k not 2 or 3, which are not semisimple or unipotent.A. Singh, M. Thakur type G 2 over fields of characteristic different from 2, for both semisimple and unipotent elements. By consulting the character table of G 2 over finite fields in [CR], one sees that reality is not true for arbitrary elements of G 2 (see also Theorem 6.11 and Theorem 6.12, in this paper). Let G be a group of type G 2 over a field k of characteristic = 2. We prove that every unipotent element in G(k) is a product of two involutions in G(k).As it turns out, the case of semisimple elements in G(k) is more delicate. We prove that a semisimple element in G(k) is real in G(k) if and only if it is a product of two involutions in G(k) (Theorem 6.3). We call a torus in G indecomposable if it can not be written as a direct product of two subtori, decomposable otherwise. We show that semisimple elements in decomposable tori are always real (Theorem 6.2). We construct examples of indecomposable tori in G containing non-real elements (Proposition 6.4 and Theorem 6.10). We work with an explicit realization of a group of type G 2 as the automorphism group of an octonion algebra. It is known (Chap. III, Prop. 5, Corollary, [Se]) that for a group G of type G 2 over k, there exists an octonion algebra C over k, unique up to a k-isomorphism, such that G ∼ = Aut(C), the group of k-algebra automorphisms of C. The group G is k-split if and only if the octonion algebra C is split, otherwise G is anisotropic and C is necessarily a division algebra. We prove that any semisimple element in G(k), either leaves invariant a quaternion subalgebra or fixes a quadraticétale subalgebra pointwise (Lemma 6.1). In the first case, reality is a consequence of a theorem of Wonenburger (Th. 4, [W1]). In the latter case, the semisimple element belongs to a subgroup SU(V, h) ⊂ G, for a hermitian space (V, h) of rank 3 over a quadratic field extension L of k, or to a subgroup SL(3) ⊂ G. We investigate these cases separately in sections 6.1 and 6.2 respectively. We discuss reality for G 2 over special fields (Proposition 6.4, Theorem 6.10 and Theorem 6.11). We show that for k with cd(k) ≤ 1 (e.g., k a finite field), every semisimple element in G(k) is a product of two involutions in G(k), and hence is real (Theorem ...

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Let k be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over k which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra A over a field k, every norm similarity of A is inner modulo scalar multiplications. It is known that k-forms of E 8 with index E 78 8,2 and anisotropic kernel a strict inner k-form of E 6 correspond bijectively (via Moufang hexagons) to Albert division algebras over k. The Kneser-Tits problem for a form of E 8 as above is equivalent to the Tits-Weiss conjecture (see [31]). Hence we provide a solution to the Kneser-Tits problem for forms of E 8 arising from pure first Tits construction Albert division algebras. As an application, we prove that for G = Aut(A), G(k)/R = 1, where A is a pure first construction Albert division algebra over k and R stands for R-equivalence in the sense of Manin ([10]).

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Let G be an algebraic group defined over a field k. We call g ∈ G real if g is conjugate to g −1 and g ∈ G(k) as k-real if g is real in G(k). An element g ∈ G is strongly real if ∃h ∈ G, h 2 = 1 (i.e. h is an involution) such that hgh −1 = g −1 . Clearly, strongly real elements are real and are product of two involutions. Let G be a connected adjoint semisimple group over a perfect field k, with −1 in the Weyl group. We prove that any strongly regular k-real element in G(k) is strongly k-real (i.e. is a product of two involutions in G(k)). For classical groups, with some mild exceptions, over an arbitrary field k of characteristic not 2, we prove that k-real semisimple elements are strongly k-real. We compute an obstruction to reality and prove some results on reality specific to fields k with cd(k) ≤ 1. Finally, we prove that in a group G of type G 2 over k, characteristic of k different from 2 and 3, any real element in G(k) is strongly k-real. This extends our results in [ST05], on reality for semisimple and unipotent real elements in groups of type G 2 .