Existence of non-standard normal Einstein metrics beyond H×K/ΔK

Ascertain whether there exist normal Einstein metrics that are not the standard Killing metric on compact homogeneous spaces outside the known class H × K/ΔK with H and K compact simple Lie groups and the specific normal metric g_b described; either construct such non-standard normal Einstein metrics or prove their nonexistence.

Background

The only known examples of normal Einstein metrics that are not the standard metric occur on spaces of the form H × K/ΔK, with a particular choice of bi-invariant inner product leading to a normal metric isometric to the Killing metric on H.

Beyond this construction, the paper proves that normal metrics are not Einstein on M = H × H/ΔK and notes broader nonexistence on aligned homogeneous spaces, leaving open whether other settings admit non-standard normal Einstein metrics.