  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.
