A-upper motives

Let E/F be a finite field extension with every intermediate field being Galois over F. Given a reductive group G over F such that G_E is of inner type, we define its A-upper motives. These motives are indecomposable and naturally related with the indecomposable summands in the motives of spectra of intermediate fields in E/F. We show that motives of projective homogeneous varieties under G are isomorphic to direct sums of Tate shifts of A-upper motives. Based on that, we get a classification of the motives of projective homogeneous varieties under absolutely simple groups of type not ^6D_4, by means of their higher Artin-Tate traces. We also show how Tits indexes over suitable field extensions determine motivic equivalence classes for these algebraic groups.