  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 $^6\!D_4$, 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.
