Let $k$ be an algebraically closed field of characteristic zero. Let $L$ be a finite-dimensional semisimple Lie algebra over $k$, and let $H \subset L$ be a Cartan subalgebra. Define the centralizer of $H$ in $L$ by
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\begin{align*}
C_L(H) := \{x \in L : [x,h] = 0 \text{ for every } h \in H\}.
\end{align*}