Intersection of Levi subgroups via intersection of their Weyl groups

Question

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 $W_M \cap W_L=\{1\}$?