 SU(2) proof of the Serre relations

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 $(\mathrm{ad}{e}_i{)}^{n_{ij}+1}{e}_j=0$.