Automorphisms of Albert algebras and a conjecture of Tits and Weiss (Q2838115)

This is a very clearly written paper proving some nice results on Albert algebras including a proof of a conjecture of Tits and Weiss. Each Albert algebra over a field arises from one of two constructions due to Tits. The author considers those produced by the first construction and not by the second. He proves that all automorphisms of such a division algebra are inner. He uses this to settle a conjecture of Tits and Weiss for the above Albert division algebras, which asserts that the structure group of the algebra is generated by the so-called \(U\)-operators and right multiplications. As this conjecture is related to the Kneser-Tits problem for groups of type \(E_8\), the author is able to answer the latter problem affirmatively for \(k\)-forms of the groups of type \(E_8\) arising from these algebras. As a further consequence, the author deduces that the R-equivalence classes are trivial for the automorphism group of an Albert division algebra obtained by Tits's first construction.