# How to find coefficients in Lie bracket relations in Cartan-Weyl basis?

For example, consider an $D_n$ Lie algebra.

The Cartan-Weyl basis satisfies the following Lie bracket relations [1, p.98]

$$\left[H^{\alpha},H^{\beta}\right]=0$$ $$\left[H^{\alpha},E^{\beta}\right]=\left(\alpha^{\vee},\beta\right)E^{\beta}$$ $$\left[E^{\alpha},E^{-\alpha}\right]=H^{\alpha}$$ $$\left[E^{\alpha},E^{\beta}\right]=e_{\alpha,\beta}E^{\alpha+\beta},\alpha\ne\beta$$ for all roots $\alpha$

I already know the set of $H^\alpha$, with $\alpha$ being the simple roots. The simple roots are also known.

I wish to determine the coefficients $e_{\alpha,\beta}$, for which it is sufficient to know the form of $E^\alpha$ for all roots $\alpha$ in a representation of $A_n$.

I would like to know if there is an method to do this.

[1] Fuchs, Jürgen, and Christoph Schweigert. Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists. Cambridge, U.K.: Cambridge University Press, 1997.

If $$-r\alpha + \beta, \ldots, \beta, \ldots, s\alpha + \beta$$ is the $\alpha$-string through $\beta$ then $e_{\alpha, \beta} = \pm(r + 1)$. There is some freedom in whether it's $(r + 1)$ or $-(r + 1)$ but it's not entirely trivial to explain. You basically get to choose the sign freely whenever $(\alpha, \beta)$ is an extraspecial pair of roots and then the signs for the remaining pairs are determined inductively according to a certain ordering on pairs (of which the extraspecial pairs are minimal).