Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $k$ of characteristic zero, and let $\mathfrak{h}$ be a Cartan subalgebra. For each $\alpha \in \mathfrak{h}^*$, define
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\begin{align*}
\mathfrak{g}_\alpha := \{x \in \mathfrak{g} : [h,x]=\alpha(h)x \text{ for every } h \in \mathfrak{h}\},
\end{align*}
Moreover, $\mathfrak{h}=\mathfrak{g}_0$, every $\operatorname{ad}(h)$ with $h\in\mathfrak{h}$ acts diagonally on $\mathfrak{g}$, and $\Phi$ spans $\mathfrak{h}^*$.