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

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 ${}^1{\mathrm{E}}_6$.

With an Appendix by Richard M.\ Weiss