Let $\mathfrak g$ be a lie algebra with a root space decomposition $\mathfrak g =\mathfrak h \oplus \oplus_{\alpha \neq 0} \mathfrak g_\alpha$ and let it have an invariant billinear form $(,)$.
Then if $e \in \mathfrak g_\alpha,\alpha \neq 0 $ and $ h \in \mathfrak h$, then $([h,e],e)=(\alpha(h)e,e)=\alpha(h)$. On the other hand, $([h,e_j],e_j)=([e_j,e_j],h)=(0,h)=0$.
Where is this contradiction coming from?