Stability of standard Einstein metrics on homogeneous spaces of non-simple Lie groups

The classification of compact homogeneous spaces of the form $M=G/K$, where $G$ is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are $4$ infinite families and $3$ isolated spaces found by Nikonorov and Rodionov in the 90s. In this paper, we prove that most of these standard Einstein metrics are unstable as critical points of the scalar curvature functional on the manifold of all unit volume $G$-invariant metrics on $M$, providing a lower bound for the coindex in the case of Ledger-Obata spaces. On the other hand, examples of stable (in particular, local maxima) invariant Einstein metrics on certain homogeneous spaces of non-simple Lie groups are also given.