Serre relations
= Serre relations
{c}
{title2=$(\operatorname{ad}e_i)^{1-a_{ij}}e_j=0$}
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 $(\operatorname{ad}e_i)^{1-a_{ij}}e_j=(\operatorname{ad}f_i)^{1-a_{ij}}f_j=0$ for $i\ne j$. The off-diagonal Cartan entries are nonpositive integers. These relations restrict which iterated simple-root brackets remain nonzero.