Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set
$$\{v \in V : \exists w \in W\ (w\cdot v = -v)\}.$$
Evidently, if $w \cdot v = -v$ and $n$ is odd, then $w^n \cdot v = -v$ as well, so it is enough to consider elements $w$ of order $2^n$. So that makes one's job easier.
I've heard tell that in fact I need only consider involutions, elements of order $2$, and that further, I can consider simple compositions of reflections in orthogonal hyperplanes. These would be useful facts to have at my disposal, but I can't immediately see why they're true. Is it the case? If so, could someone point me to a proof?
(In case it helps, the case I really care about is the Coxeter group $W(E_6)$ of type $E_6$ acting as the Weyl group of the Lie group $E_6$ and the action on its Cartan algebra.)