On the Tits–Weiss conjecture and the Kneser–Tits conjecture for $\mathrm {E}^{78}_{7,1}$ and $\mathrm {E}^{78}_{8,2}$ (With an Appendix by R. M. Weiss)

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 $^1\mathrm {E}_6$ .

Keywords

• 20G41
• 17C40
• 20G15