Suppose I have a root system $\Phi$ (of a semisimple Lie algebra) with a set of simple roots $\Delta$. I am interested in describing Weyl group elements $w$ preserving a given subset $\Delta'$ in the sense that $w(\Delta')=\Delta'$.

Denote by $\Delta''$ the largest subset of $\Delta$ such that $\Delta'\cap\Delta''=\emptyset$ and all roots in $\Delta'$ are orthogonal to roots in $\Delta''$. Obviously, any product of simple reflections corresponding to roots of $\Delta''$ fixes $\Delta'$. I wonder, if this gives me all such $w$.

As far as I understand, my question is equivalent to asking which Weyl group elements fix a given Levi subalgebra.

Suppose I have a root system $\Phi$ (of a semisimple Lie algebra) with a set of simple roots $\Delta$. I am interested in describing Weyl group elements $w$ preserving a given subset $\Delta'$ in the sense that $w(\Delta')=\Delta'$.

Denote by $\Delta''$ the largest subset of $\Delta$ such that $\Delta'\cap\Delta''=\emptyset$ and all roots in $\Delta'$ are orthogonal to roots in $\Delta''$. Obviously, any product of simple reflections corresponding to roots of $\Delta''$ fixes $\Delta'$. For example, one may hope that this gives me all such $w$ if $\Delta'$ is a connected subset of $\Delta$.

As far as I understand, my question is equivalent to asking which Weyl group elements fix a given Levi subalgebra.