  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.
