Let $G$ be an exceptional simple algebraic group over a field $k$, and let $X$ be a projective $G$-homogeneous variety such that $G$ splits over $k(X)$. We classify such varieties $X$. This classification allows us to relate the Rost invariant of groups of type $E_7$ and their isotropy and to give a two-line proof of the triviality of the kernel of the Rost invariant for such groups. Apart from this, it plays a crucial role in the solution of a problem posed by Serre for groups of type $E_8$