Weyl groups of Levi factors and their cosets

Let $G$ be a connected semisimple algebraic group defined over an algebraically closed field $F$ of characteristic $0$. Let $B$ be a minimal parabolic subgroup of $G$ (i.e. a Borel subgroup), let $P$ be a parabolic subgroup of $G$ containing $B$, and let $L$ be a Levi factor of $P$. Letting $\pi\colon P \rightarrow L$ be the projection, define $B_L = \pi(B)$.

Letting $W$ be the Weyl group of $G$, define

$$W^L = \{\text{$w \in W$ $|$ $w B_L w^{-1} \subset B$}\}.$$

It is clear that this is well-defined. Letting $W_L$ be the Weyl group of $L$, a paper I am reading claims that $W^L$ is a set of coset representatives for $W/W_L$. I can verify this for many examples (for instance, general linear and symplectic groups), but I don’t see how to do it in general. Can someone either explain this or give a reference?

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Source : Link , Question Author : Sarah , Answer Author : Community