Orbits of subsystem stabilisers

Let $\Phi$ be a reduced irreducible root system. We consider pairs $(S,X(S))$, where $S$ is a closed set of roots, $X(S)$ is its stabiliser in the Weyl group $W(\Phi)$. We are interested in such pairs maximal with respеct to the following order: $(S_1,X(S_1))\le (S_2,X(S_2))$ if $S_1\subseteq S_2$ and $X(S_1)\le X(S_2)$. Main theorem asserts that if $\Delta$ is a root subsystem such that $(\Delta,X(\Delta))$ is maximal with respect to the above order, then $X(\Delta)$ acts transitively both on the long and short roots in $\Phi\setminus\Delta$. This result is a broad generalisation of the transitivity of the Weyl group on roots of given length.