Identity for the $N_{\alpha \beta}$ of a Lie algebra from Killing form

Question

Let $\alpha$, $\beta$ be two roots of a general Lie algebra of rank $r$, and $e_\alpha$ and $e_\beta$ their respective eigenvectors, with $$[e_\alpha,e_\beta]=N_{\alpha,\beta}e_{\alpha+\beta}.$$

Let the Killing form be normalised such that $$B(e_\alpha,e_\beta)=\delta_{\alpha,-\beta}.$$ How can I prove that $$N_{\alpha,\beta}=N_{-\alpha-\beta,\beta},$$ using the invariance of the Killing form?

Answer 1

You should be able to prove that by computing

$$B([e_{-\alpha-\beta},e_{\beta}],e_{\alpha})$$

in two different ways; however, when I do that, I get $N_{\alpha,\beta}=\color{red}{-}N_{-\alpha-\beta,\beta}$.