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

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For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map RG: H¹(F, G) --> H³(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 RG has trivial kernel if G is quasi-split of type E₆ or E₇. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.