# Intersection of Levi subgroups via intersection of their Weyl groups

Let $$G$$ be a connected reductive group over $$\mathbb{C}$$. We fix a maximal torus $$T \subset G$$. Let $$M,L \subset G$$ be its Levi subgroups containing $$T$$ (note that we do note assume that $$M,L$$ are standard with respect to the same Borel subgroup). Let $$\Delta$$ be the set of roots of $$(G,T)$$ and $$W$$ is the corresponding Weyl group. Let $$\Delta_M,\Delta_L\subset \Delta$$ be the roots of $$M,L$$ respectively, $$W_M,W_L\subset W$$ are their Weyl groups. Assume that $$\Delta_M\cap \Delta_L=\varnothing$$, does this imply that $$W_M\cap W_L=\{1\}$$?