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We prove that the structure group of any Albert algebra over an arbitraryfield is $R$-trivial. This implies the Tits-Weiss conjecture for Albertalgebras 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 showthat some standard conjectures on the groups of $R$-equivalence classes inalgebraic groups and the norm principle are true for strongly inner forms oftype $^1\mathrm{E}_6$.

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