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Let p be a positive prime number and X be a Severi-Brauer variety of acentral division algebra D of degree p^n, with n>0. We describe all shifts ofthe motive of X in the complete motivic decomposition of a variety Y, whichsplits over the function field of X and satisfies the nilpotence principle. Inparticular, we prove the motivic decomposability of generalized Severi-Brauervarieties X(p^m,D) of right ideals in D of reduced dimension p^m,m=0,1,...,n-1, except the cases p=2, m=1 and m=0 (for any prime p), wheremotivic indecomposability was proven by Nikita Karpenko.

Decompositions of motives of generalized Severi-Brauer varieties