$R$-triviality of Groups of Type ${\mathrm{F}}_4$ Arising from the First Tits Construction

Abstract

Any group of type ${\mathrm{F}}_4$ is obtained as the automorphism group of an Albert algebra. We prove that such a group is $R$-trivial whenever the Albert algebra is obtained from the first Tits construction. Our proof uses cohomological techniques and the corresponding result on the structure group of such Albert algebras.