On the Tits-Weiss Conjecture and the Kneser-Tits Conjecture for $\mathrm{E}^{78}_{7,1}$ and $\mathrm{E}^{78}_{8,2}$

Abstract

We prove that the structure group of any Albert algebra over an arbitrary field is $R$-trivial. This implies the Tits-Weiss conjecture for Albert algebras and the Kneser-Tits conjecture for isotropic groups of type $\mathrm{E}_{7,1}^{78}, \mathrm{E}_{8,2}^{78}$. As a further corollary, we show that some standard conjectures on the groups of $R$-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type $^1\mathrm{E}_6$.