Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T \subset G$. Let $M,L \subset G$ be two connected reductive subgroups containing $T$. Let $\Delta$ be the root system of $G$ with respect to $T$, and let $W$ denote the corresponding Weyl group. Let $\Delta_M,\Delta_L\subset \Delta$ be the root systems of $M,L$, respectively (again with respect to $T$), and let $W_M,W_L\subset W$ denote their Weyl groups. Assume that $\Delta_M\cap \Delta_L=\varnothing$; does this imply that $W_M\cap W_L=\{1\}$?