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.