I'm having difficulty with this result (given as 2 lines in my book):

Let $\Phi$ be a root system as defined <http://en.wikipedia.org/wiki/Root_system> and let $W$ be a group generated by the reflections $s\_\alpha$ for $\alpha \in \Phi$.

Suppose that we know each element of $W$ fixes pointwise the orthogonal complement, $U^\perp$ of the subspace spanned by $\Phi$, $U$.

Somehow, this implies that only $e$ can fix $\Phi$.

I can kinda see how we would want this to be true, but I'm unsure how to prove it.