  We compute the invariants so of Weyl groups in mod 2 Milnor K-theory and more
general cycle modules, which are annihilated by 2. Over a base field of
characteristic coprime to the group order, the invariants decompose as direct
sums of the coefficient module. All basis elements are induced either by
Stiefel-Whitney classes or specific invariants in the Witt ring. The proof is
based on Serre's splitting principle that guarantees detection of invariants on
elementary abelian 2-subgroups generated by reflections.
