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 $\mathrm{E}_7$ 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 $\mathrm{E}_8$