Invariant metrics on homogeneous spaces with equivalent isotropy summands

The space of $G$-invariant metrics on a homogeneous space $G/H$ is in one-to-one correspondence with the set of inner products on the tangent space $\fr{m}\cong T_{{\it o}}(G/H)$, which are invariant under the isotropy representation. When all the isotropy summands are inequivalent to each other, then the metric is called diagonal. We will describe a special class of $G$-invariant metrics %with additional symmetries. in the case where the isotropy representation of $G/H$ contains some equivalent isotropy summands. Even though this problem has been considered sporadically in the bibliography, in the present article we provide a more systematic and organized description of such metrics. This will enable us to simplify the problem of finding $G$-invariant Einstein metrics for homogeneous spaces. We also provide some applications.