SU(2) proof of the Serre relations
= SU(2) proof of the Serre relations
{c}
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 $n_{ij}\ge0$ integral and make $(\operatorname{ad}e_i)^{n_{ij}+1}e_j=0$.