Serre relations ($(\mathrm{ad}{e}_i{)}^{1-a_{ij}}{e}_j=0$)

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