it is proposition 14.31 in Fulton-Harris book. The proof goes like this. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, and assume $\mathfrak{z}\subseteq\mathfrak{h}^*$ were preserved by the action of the Weyl group. Let $\mathfrak{g}'$ be the subspace spanned by the subalgebras $\{\mathfrak{s}_{\alpha}\}_{\alpha\in\mathfrak{z}}$ ($s_{\alpha}$ is the subalgebra isomorphic to $sl_2(\mathbb{C})$ corresponding to the root $\alpha$). Then it shows that $\mathfrak{g}'$ is an ideal. Then it says
Thus either all the roots lie in $\mathfrak{h}$ or all the roots are perpendicular to $\mathfrak{h}$
I don't understand this sentence. Why is it true?
Thanks.