Chevalley basis

A Chevalley basis of a complex semisimple Lie algebra consists of suitable coroot elements and one vector $e_{\alpha }$ in each root space, chosen so all Lie bracket structure constants are integers. In particular $[e_{\alpha },e_{\beta }]=N_{\alpha ,\beta }{e}_{\alpha +\beta }$ when the sum is a root, with $N_{\alpha ,\beta }=\pm (p+1)$ where $\beta -p\alpha$ is the last negative-direction root in the root string. For independent roots with no root sum the bracket is zero. The integral lattice makes reduction to other fields possible.

For simple-root generators $e_i,f_i,h_i$ with $[h_i,e_j]=a_{ij}{e}_j$, the Serre relations include $[e_i,f_j]={\delta }_{ij}{h}_i$ and $(\mathrm{ad}{e}_i{)}^{1-a_{ij}}{e}_j=(\mathrm{ad}{f}_i{)}^{1-a_{ij}}{f}_j=0$ for $i\ne j$. The off-diagonal Cartan entries are nonpositive integers. These relations restrict which iterated simple-root brackets remain nonzero.

For distinct simple roots, ${\alpha }_j-{\alpha }_i$ has mixed-sign simple-root coefficients and is not a root. Thus $[f_i,e_j]=0$. In the finite-dimensional adjoint representation of the simple-root SU(2) subalgebra, $e_j$ is a lowest-weight vector of weight $a_{ij}$. Standard SU(2) ladders force $a_{ij}=-n_{ij}$ with $n_{ij}\ge 0$ integral and make $(\mathrm{ad}{e}_i{)}^{n_{ij}+1}{e}_j=0$.

For equal-length roots $r,s$ with a root sum, their angle is $2\pi /3$. The vectors $2r+s$ and $r+2s$ have squared length three times a root's squared length, so neither is a root. Thus $[e_r,e_{r+s}]=[e_s,e_{r+s}]=0$. Apply the central-commutator exponential identity to $A=t\mathrm{ad}{e}_r$ and $B=u\mathrm{ad}{e}_s$, whose commutator is $tu{N}_{r,s}\mathrm{ad}{e}_{r+s}$, to obtain the displayed identity over the complex numbers.