Automorphisms of Albert algebras and a conjecture of Tits and Weiss

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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 E₈ with index E⁷⁸₈,₂ and anisotropic kernel a strict inner k-form of E₆ correspond bijectively (via Moufang hexagons) to Albert division algebras over k. The Kneser-Tits problem for a form of E₈ as above is equivalent to the Tits-Weiss conjecture (see TW). Hence we provide a solution to the Kneser-Tits problem for forms of E₈ 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 (M).