Compact weakly symmetric spaces and spherical pairs

Let $(G,H)$ be a spherical pair and assume that $G$ is a connected compact simple Lie group and $H$ a closed subgroup of $G$. We prove in this paper that the homogeneous manifold $G/H$ is weakly symmetric with respect to $G$ and possibly an additional fixed isometry $\mu$. It follows that M. Krämer's classification list of such spherical pairs also becomes a classification list of compact weakly symmetric spaces. In fact, our proof involves a case-by-case study of the isotropy representations of all spherical pairs on Krämer's list.