On computing the spherical roots for a class of spherical subgroups

Given a connected reductive algebraic group $G$, we consider the class of spherical subgroups $H \subset G$ such that $H$ is regularly embedded in a parabolic subgroup $P \subset G$ and $H,P$ have a common Levi subgroup $L$. In a previous paper, the author developed a fast algorithm that reduces the computation of the set of spherical roots for such subgroups $H$ to the case where the quotient of Lie algebras $\operatorname{Lie} P / \operatorname{Lie} H$ is a strictly indecomposable spherical $L$-module. In this paper, we complete the classification of all such cases and compute the spherical roots for each of them, which enables one to use the above fast algorithm directly for computing the spherical roots for arbitrary spherical subgroups in the class under consideration.