On the splitting principle for cohomoligical invariants of reflection groups

Let k0 be a field of characteristic not two, (V, b) a finite-dimensional regular bilinear space over k0, and W a subgroup of the orthogonal group of (V, b) with the property that the subring of W-invariants of the symmetric algebra of V is a polynomial algebra over k0. We prove that Serre’s splitting principle holds for cohomological invariants of W with values in Rost’s cycle modules.