If you have an algebraic group $G$ with a split $BN$-pair and Weyl group $W$, the Chevalley commutator relations state there is a total ordering on the roots $\Phi$ such that for $\alpha,\beta\in\Phi$ with $\beta\neq\pm\alpha$, and $X_\alpha$ the root subgroup for $\alpha$, $$ [X_\alpha,X_\beta]\subseteq\prod_{\gamma=i\alpha+j\beta,\ i,j>0,\ \alpha<\gamma,\beta<\gamma}X_\gamma. $$
Is there any partial result, or special cases where one can determine where $[X_\alpha,X_{-\alpha}]$ lives, as in if it's contained in any particular subgroup smaller than $G$ itself?