Let $R$ be the root system of a simple complex Lie algebra $g$ with respect to some Cartan subalgebra $h$. Chevalley showed there is a basis of $g$ given by the simple coroots {$H_{\alpha_i}=\alpha_i^\vee\in h$} and root vectors $X_\alpha\in g_\alpha$ for each $\alpha\in R$. This basis has the following properties:

$[H_{\alpha_i},H_{\alpha_j}]=0$

$[H_{\alpha_i},X_\beta]=\beta(H_{\alpha_i})X_\beta$

$[X_{\alpha},X_{-\alpha}]=H_\alpha=\alpha^\vee\in h$

($\ast$) $[X_\alpha,X_\beta]=\pm(p+1)X_{\alpha+\beta}$, when $\alpha+\beta\in R$ and $p$ is the greatest positive integer such that $\beta-p\alpha\in R$. Otherwise, if $\alpha+\beta$ is not a root, then the bracket is zero.

References for this can be found in Serre's book on semisimple complex Lie algebras or Humphrey's book or Wikipedia.

Does anybody know a simple way to determine the sign $\pm$ in the fourth property ($\ast$)?

I cannot find a reference and my French is not good, so reading the original works by Chevalley and Tits isn't a viable option. In particular, I need to find a sign convention that will work for $g$ of type $F_4$.

Thanks so much.

) with your favourite signs, and then work out the rest of () with a lot of patience.) – fherzig Dec 17 '11 at 23:16