# Chevalley basis

The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives $\pm\alpha_i$. The relations among the generators are the following:
$[H_{\alpha_i},H_{\alpha_j}]=0$
$[H_{\alpha_i},E_{\alpha_j}]=\alpha_j(H_{\alpha_i}) E_{\alpha_j}$
$[E_{-\alpha_i},E_{\alpha_i}] = H_{\alpha_i}$
$[E_{\beta},E_{\gamma}]=\pm(p+1)E_{\beta+\gamma}$
where the last relation is imposed only if $\beta + \gamma$ is a root and p is the greatest positive integer such that γ − pβ is a root. The sign in the relation can be fixed arbitrarily.