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. Let
be the corresponding root-space decomposition, where
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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*}
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Let $\kappa:\mathfrak{g}\times\mathfrak{g}\to k$ be the Killing form. For each root $\alpha\in\Phi$, let $t_{\alpha}\in\mathfrak{h}$ be the unique element satisfying