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On the Tits-Weiss Conjecture and the Kneser-Tits Conjecture for $\mathrm{E}^{78}_{7,1}$ and $\mathrm{E}^{78}_{8,2}$ (1911.12908v1)
Published 29 Nov 2019 in math.RA, math.AG, and math.GR
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 $1\mathrm{E}_6$.