  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 $|W|$. We then show
that this principle for such groups holds also for Witt- and Milnor-Witt
$K$-theory invariants.
