Let $\mathfrak{g}$ be the lie algebra of a compact lie group, so that the Killing form $(,)$ is negative definite. Let the root space decomposition $\mathfrak g =\mathfrak h \oplus \oplus_{\alpha \neq 0} \mathfrak g_\alpha$ be given.
If $e \in \mathfrak{g_\alpha}$ for $\alpha \neq 0$, then $\alpha(h)(e,e)=([h,e],e)=([e,h],e)=-\alpha(h)(e,e)$ and $(e,e)=0$. This contradicts the form being negative definite.
Where is this contradiction coming from?
One does not have a root space decomposition when $\mathfrak g$ is the lie algebra of a compact lie group. This is because any complex lie group is necessarily not compact.