Proposition 2

If W is irreducible and h is even, the unique element of W that transforms C to −C is $c^{h/2}$.

We use the notation in the proof of Th. 1. Since $c|P$ is a rotation through an angle $\frac{2\pi}{h}$, $c^{h/2}$ transforms $z'$ to $-z'$, $z''$ to $-z''$, and hence $z' + z'' = z$ to $-z$. Now $z \in C$, so the chamber $c^{h/2}(C)$ is necessarily $-C$.