Stability of non-diagonal Einstein metrics on homogeneous spaces $H\times H\mathrm{/}\mathrm{\Delta }K$

We consider the homogeneous space $M=H\times H\mathrm{/}\mathrm{\Delta }K$, where $H\mathrm{/}K$ is an irreducible symmetric space and $\mathrm{\Delta }K$ denotes diagonal embedding. Recently, Lauret and Will provided a complete classification of $H\times H$-invariant Einstein metrics on M. They obtained that there is always at least one non-diagonal Einstein metric on $M$, and in some cases, diagonal Einstein metrics also exist. We give a formula for the scalar curvature of a subset of $H\times H$-invariant metrics and study the stability of non-diagonal Einstein metrics on $M$ with respect to the Hilbert action, obtaining that these metrics are unstable with different coindexes for all homogeneous spaces $M$.