Let $k$ be an algebraically closed field of characteristic $0$. Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $k$, and let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For each linear functional $\alpha \in \mathfrak{h}^*$, define the simultaneous eigenspace
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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, $\mathfrak{g}_0 = \mathfrak{h}$, each operator $\operatorname{ad}(h):\mathfrak{g}\to\mathfrak{g}$ with $h \in \mathfrak{h}$ is diagonalizable over $k$, and $\Phi$ spans $\mathfrak{h}^*$.