Let $L$ be a nonzero finite-dimensional semisimple Lie algebra over an algebraically closed field $k$ of characteristic zero, let $H \subset L$ be a Cartan subalgebra, and let
\begin{align*}
L_\alpha = \{x \in L : [h,x] = \alpha(h)x \text{ for every } h \in H\}.
\end{align*}
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Equip the real [vector space](/page/Vector%20Space) $\operatorname{span}_{\mathbb{R}}\Phi$ with the symmetric [bilinear form](/page/Bilinear%20Form) induced by the Killing form of $L$. Then $L$ is simple if and only if $\Phi$ is irreducible.