Chevalley basis

Remarks

From these rules, $[h_i, h_j] = 0$, $[h_i, x_\alpha] = \alpha(h_i) x_\alpha$ with $\alpha(h_i) \in \Z$, each $h_\alpha$ is an integer combination of the $h_i$, and $[x_\alpha, x_\beta] = 0$ if $\alpha + \beta \notin \Phi \cup \set{0}$. So all structure constants are integers. Example: in $\mathfrak{sl}_2(\C)$, the basis $h = \operatorname{diag}(1, -1)$, $x_\alpha = E_{12}$, $x_{-\alpha} = E_{21}$ is a Chevalley basis, since $[E_{12}, E_{21}] = h$ and $\alpha(h) = 2$; clause 3 is empty. Non-example: the basis $h, 2E_{12}, E_{21}$ fails clause 2, since $[2E_{12}, E_{21}] = 2h \ne h_\alpha$. In $\mathfrak{sl}_n(\C)$, the matrix units $E_{ij}$, $i \ne j$, with $h_i = E_{ii} - E_{i+1, i+1}$ form a Chevalley basis. By a theorem of Chevalley, every complex semisimple Lie algebra has a Chevalley basis. The $\Z$-span of a Chevalley basis is a Lie ring, from which the Chevalley groups over any field are built.