Classification of standard Einstein metrics for non-simple H

Classify all standard Einstein metrics—those defined by the negative Killing form—on compact homogeneous spaces H/K where H is a non-simple compact Lie group (i.e., H has more than one simple factor).

Background

For H simple, Wang and Ziller classified compact homogeneous spaces H/K whose standard metric (induced by −Kill(H) on the isotropy) is Einstein, yielding many families and examples.

The authors note that while much is known in the simple case, extending the classification to non-simple H remains an outstanding problem.