My paper Shells of twisted flag varieties and the Rost invariant, joint with Victor Petrov and Nikita Semenov, will appear in Duke Mathematical Journal. We introduce new techniques for studying the Krull-Schmidt decomposition of mod p Chow motives of projective homogeneous varieties, by generalizing Vishik’s notion of shells on quadrics to arbitrary projective homogeneous varieties. We then apply our techniques to calculate some decompositions, relate decompositions to invariants of the group, and relate isotropy of the group to invariants of the group by way of the decompositions. Along the way we settle a 20-year-old question about isotropy of groups of type E7 and prove analogous results for E8.