For G a simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map $ R_G : H^1(F, G) \rightarrow H^3(F, \mathbb{Q} / \mathbb{Z}(2)) $
 . This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for exceptional Jordan algebras as special cases. We show that R
 G
 has trivial kernel if G is quasi-split of type E
 6 or E
 7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.

/pdf/the-rost-invariant-has-trivial-kernel-for-quasi-split-groups-1gybf2uh08.pdf

The rost invariant has trivial kernel for quasi-split groups of low rank

For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map R_G: H^1(F, G) --> H^3(F, Q/Z(2)). This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for Albert algebras as special cases. We show that R_G has trivial kernel if G is quasi-split of type E_6 or E_7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.

The Rost invariant has trivial kernel for quasi-split groups of low rank

Simple structurable algebras of skew-dimension one

It is well-known that every algebraic group of type F_4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^1E_6 with trivial Tits algebras arise as the isometry groups of norm forms of such Jordan algebras. We describe a similar relationship between groups of type E_6 and groups of type E_7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E_7 with trivial Tits algebras. The underlying algebraic structure for the relationship considered here are a sort of 56-dimensional structurable algebra which are forms of an algebra constructed from an exceptional Jordan algebra.

Structurable algebras and groups of type E_6 and E_7

Structurable H∗-algebras

Isotopes of some nonassociative algebras with involution

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Isotopes of some nonassociative algebras with involution