  Let G be a linear algebraic group over a field F and X be a projective
homogeneous G-variety such that G splits over the function field of X. In the
present paper we introduce an invariant of G called J-invariant which
characterizes the motivic behaviour of X. This generalizes the respective
notion invented by A. Vishik in the context of quadratic forms. As a main
application we obtain a uniform proof of all known motivic decompositions of
generically split projective homogeneous varieties (Severi-Brauer varieties,
Pfister quadrics, maximal orthogonal Grassmannians, G2- and F4-varieties) as
well as provide new examples (exceptional varieties of types E6, E7 and E8). We
also discuss relations with torsion indices, canonical dimensions and
cohomological invariants of the group G.
