  In the present article we investigate ordinary and equivariant Rost motives.
We provide an equivariant motivic decomposition of the variety X of full flags
of a split semisimple algebraic group over a smooth base scheme, study torsion
subgroup of the Chow group of twisted forms of X, define some equivariant Rost
motives over a field and ordinary Rost motives over a general base scheme, and
relate equivariant Rost motives with classifying spaces of some algebraic
groups.
