AUTOMORPHISMS OF ALBERT ALGEBRAS AND A CONJECTURE OF TITS AND WEISS II.

Let G be a simple, simply connected algebraic group with Tits index E⁷⁸ _8,2 or E⁷⁸ _7,1, defined over a field k of arbitrary characteristic. We prove that there exists a quadratic extension K of k such that G is R-trivial over K; i.e., for any extension F of K, G(F)/R = {1}, where G(F)/R denotes the group of R-equivalence classes in G(F), in the sense of Manin. As a consequence, it follows that the variety G is retract K-rational and that the Kneser-Tits conjecture holds for these groups over K. Moreover, G(L) is projectively simple as an abstract group for any field extension L of K. In their monograph, J. Tits and Richard Weiss conjectured that for an Albert division algebra A over a field k, its structure group Str(A) is generated by scalar homotheties and its U-operators. This is known to be equivalent to the Kneser-Tits conjecture for groups with Tits index E⁷⁸ _8,2. We settle this conjecture for Albert division algebras which are first constructions, in the affirmative. These results are obtained as corollaries to the main result, which shows that if A is an Albert division algebra which is a first construction and Γ its structure group, i.e., the algebraic group of the norm similarities of A, then Γ(F)/R = {1} for any field extension F of k; i.e., Γ is R-trivial. [ABSTRACT FROM AUTHOR]