On the splitting principle for cohomological invariants of reflection groups

Let ${\mathrm{k}}_0$ be a field and $W$ a finite orthogonal reflection group over ${\mathrm{k}}_0$. We prove Serre’s splitting principle for cohomological invariants of $W$ with values in Rost’s cycle modules (over ${\mathrm{k}}_0$) if the characteristic of ${\mathrm{k}}_0$ is coprime to $\mathrm{\mid }W\mathrm{\mid }$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants.