Automorphisms of Albert algebras and a conjecture of Tits and Weiss II

On $R$-triviality of $F_4$-II

“…We have, Proposition 3.2 (Prop. 3.2, [25]). Let A = J(B, σ, u, µ) be an Albert algebra over k with S := (B, σ) + a division algebra and K = Z(B).…”

“…All base fields will be assumed to be infinite of arbitrary characteristic unless specified otherwise. The matrial below is being reproduced from ( [25]) for convenience. For this, we refer to ( [20] or [18]).…”

“…We continue to set up notation for the rest of the paper, again mostly following ( [25]). Let A be an Albert algebra over k and S ⊂ A a subalgebra.…”

“…These groups have the structure group of an Albert division algebra as their anisotropic kernel. In ( [24]) we had proved R-triviality for Aut(A) for pure first Tits construction Albert algebras, while in ( [25]), R-triviality of the structure group was proved to settle the Tits-Weiss conjecture for first Tits construction Albert division algebras, as well as reduced Albert algebras. In the paper ( [14]), a proof of Kneser-Tits conjecture for groups of type E 66 8,2 is given by proving R-triviality of such groups.…”

R‐triviality of groups of type F4 arising from the first Tits construction

“…Proof Let be the corresponding Albert algebra. By [23, 24], the structure group is ‐trivial. Hence the assertion follows from the above corollary.…”