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

TL;DR

This paper proves the Tits-Weiss and Kneser-Tits conjectures for certain algebraic groups related to Albert algebras and exceptional groups of types E7 and E8, confirming key conjectures in algebraic group theory.

Contribution

It establishes the $R$-triviality of structure groups of Albert algebras over any field, proving the Tits-Weiss and Kneser-Tits conjectures for specific types of algebraic groups.

Findings

Proves $R$-triviality of structure groups of Albert algebras.

Confirms the Tits-Weiss conjecture for Albert algebras.

Validates the Kneser-Tits conjecture for certain isotropic groups of types E7 and E8.

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 $^1\mathrm{E}_6$.