Chow motives and the Rost invariant

The Rost invariant is an invariant of degree 3 of linear algebraic groups. Its existence was conjectured by Serre and proved by Rost in the beginning of 90s. In his paper "Cohomologie galoisienne" Serre showed that the coprime components of the Rost invariant of any group of type F4 over a purely transcendental extension of degree 1 of a p-adic field satisfy a non-trivial relation. Then he asked about other relations existing between coprime components. In the talk I will give another example of such relation. To do this I explain first two new general methods to compute motives of homogeneous varieties. These methods generalize results of Chernousov, Merkurjev, and Vishik. The subject of my talk is joint work with Garibaldi and Petrov.