On $R$-triviality of $F_4$

Abstract

It is known that simple algebraic groups of type $F_4$ defined over a field $k$ are precisely the full automorphism groups of Albert algebras over $k$. We explore $R$-triviality for the group $\text{\bf Aut}(A)$ when $A$ is an Albert algebra. In this paper, we consider the case when $A$ is an Albert division algebra, that arises from the first Tits construction. We prove that $\text{\bf Aut}(A)$ is $R$-trivial, in the sense of Manin.