  We establish the complete classification of Chow motives of projective
homogeneous varieties for $p$-inner semi-simple algebraic groups, with
coefficients in $\mathbb{Z}/p\mathbb{Z}$. Our results involve a new motivic
invariant, the Tate trace of a motive, defined as a pure Tate summand of
maximal rank. They apply more generally to objects of the Tate subcategory
generated by upper motives of irreducible, geometrically split varieties
satisfying the nilpotence principle. Using Chernousov-Gille-Merkurjev
decompositions and their interpretation through
Bialynicki-Birula-Hesselink-Iversen filtrations due to Brosnan, we then
generalize the characterization of the motivic equivalence of inner semi-simple
groups through the higher Tits $p$-indexes. We also define the motivic
splitting pattern and the motivic splitting towers of a summand of the motive
of a projective homogeneous variety, which correspond for quadrics to the
classical splitting pattern and Knebusch tower of the underlying quadratic
form.
