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.
Best Answer
There is a good discussion of these issues in the paper of A. Cohen, S. Murray and D.E. Taylor, "Computing in groups of Lie type", Math. Comp. 73, Number 247, 1477–1498, (2003), especially section 3 (referring to earlier work, e.g., of Carter). They explain in particular how the signs can be all reduced to so-called "extraspecial pairs", which can be chosen arbitrarily.
In Magma at least, one can see which extraspecial signs have been chosen using the "ExtraspecialSigns" command. For instance, one can see using this that GAP and Magma use (or used, I haven't checked the latest versions...) different constants for E_8.