Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $\mathbb{C}$, let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra, and let $\kappa: \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}$ denote the Killing form. For each root $\alpha \in \mathfrak{h}^*$, define the root space
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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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If $\alpha,\beta \in \mathfrak{h}^*$ are roots and $\alpha+\beta \ne 0$, then
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\begin{align*}
\kappa(x,y)=0
\end{align*}
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for every $x \in \mathfrak{g}_{\alpha}$ and every $y \in \mathfrak{g}_{\beta}$. Equivalently,