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

Question

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.

Answer 1

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).

The book you want to look at is Simple Groups of Lie Type by Carter, specifically Proposition 4.2.2 and the definition of extraspecial pairs is in the paragraph above that.