Automorphisms of Albert algebras and a conjecture of Tits and Weiss II

Abstract

Let $G$ be a simple, simply connected algebraic group with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary characteristic. We prove that there exists a quadratic extension $K$ of $k$ such that $G$ is $R$-trivial over $K$; i.e., for any extension $F$ of $K$, $G(F)/R=\{1\}$, where $G(F)/R$ denotes the group of $R$-equivalence classes in $G(F)$, in the sense of Manin. As a consequence, it follows that the variety $G$ is retract $K$-rational and that the Kneser–Tits conjecture holds for these groups over $K$. Moreover, $G(L)$ is projectively simple as an abstract group for any field extension $L$ of $K$. In their monograph, J. Tits and Richard Weiss conjectured that for an Albert division algebra $A$ over a field $k$, its structure group $Str(A)$ is generated by scalar homotheties and its $U$-operators. This is known to be equivalent to the Kneser–Tits conjecture for groups with Tits index $E_{8,2}^{78}$. We settle this conjecture for Albert division algebras which are first constructions, in the affirmative. These results are obtained as corollaries to the main result, which shows that if $A$ is an Albert division algebra which is a first construction and $\Gamma$ its structure group, i.e., the algebraic group of the norm similarities of $A$, then $\Gamma (F)/R=\{1\}$ for any field extension $F$ of $k$; i.e., $\Gamma$ is $R$-trivial.