Automorphisms of Albert algebras and a conjecture of Tits and Weiss

Abstract

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 $E_{8,2}^{78}$ 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 \cite{TW}). 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 $G=\mathbf{Aut}(A),G(k)/R=1$, where $A$ is a pure first construction Albert division algebra over $k$ and $R$ stands for $R$-equivalence in the sense of Manin (\cite{M}).

41 pages