Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $k$ of characteristic $0$, let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra, let $\Phi \subset \mathfrak{h}^*$ be the corresponding root system, and let
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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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denote the root space for $\alpha \in \Phi$. Let $\kappa: \mathfrak{g} \times \mathfrak{g} \to k$ be the Killing form. For each $\alpha \in \Phi$, let $t_\alpha \in \mathfrak{h}$ be the unique element satisfying