  Given any compact homogeneous space $H/K$ with $H$ simple, we consider the
new space $M=H\times H/\Delta K$, where $\Delta K$ denotes diagonal embedding,
and study the existence, classification and stability of $H\times H$-invariant
Einstein metrics on $M$, as a first step into the largely unexplored case of
homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein
metrics on $M$ for most spaces $H/K$ such that their standard metric is
Einstein (e.g., isotropy irreducible) and the Killing form of $\mathfrak{k}$ is
a multiple of the Killing form of $\mathfrak{h}$ (e.g., $K$ simple), a class
which contains $17$ families and $50$ individual examples. A complete
classification is obtained in the case when $H/K$ is an irreducible symmetric
space. We also study the behavior of the scalar curvature function on the space
of all normal metrics on $M=H\times H/\Delta K$ (none of which is Einstein),
obtaining that the standard metric is a global minimum.
