Existence and uniqueness of normal Einstein metrics on homogeneous spaces with non-simple Lie groups

Determine whether normal Einstein metrics exist and whether they are unique on compact homogeneous spaces M = G/K where the acting Lie group G is non-simple (i.e., has at least two simple factors), restricting attention to the class of normal metrics induced by bi-invariant inner products on the Lie algebra g of G.

Background

When the transitive Lie group G of a compact homogeneous space M = G/K is not simple, the space of G-invariant metrics contains the s-parameter subspace of normal metrics, defined via bi-invariant inner products on the Lie algebra g. This introduces new structure beyond the well-studied simple-group case.

The paper shows, for the specific family M = H × H/ΔK with H compact simple, that normal metrics are never Einstein; nonetheless, the general existence and uniqueness questions for normal Einstein metrics in the broader non-simple setting remain open.