Motivic decomposition of anisotropic varieties of type F₄ into generalized Rost motives

This an extended version of the previous preprint dated by February 2005. We prove that the Chow motive of an anisotropic projective homogeneous variety of type F4 is isomorphic to the direct sum of twisted copies of a generalized Rost motive. In particular, we provide an explicit construction of a generalized Rost motive for a generically splitting variety for a symbol in K_3^M(k)/3. We also establish a motivic isomorphism between two anisotropic non-isomorphic projective homogeneous varieties of type F4. All our results hold for Chow motives with integral coefficients.