Subspace invariant under irreducible Coxeter group

Question

I'm trying to show that if $G$ is an irreducible Coxeter group, then it acts irreducibly on vector space $V$. That is, $V$ has no nontrivial $G$-invariant subspaces. I started by assuming that $V$ has a $G$-invariant subspace $W$. Then its orthogonal complement $W'$ is also $G$-invariant. If I can show that any root of $G$ lies either in $W$ or in $W'$, then the rest of the proof follows easily. How can I know this is true?