Einstein metrics on aligned homogeneous spaces with two factors

Abstract Given two homogeneous spaces of the form and , where and are compact simple Lie groups, we study the existence problem for ‐invariant Einstein metrics on the homogeneous space . For the large subclass of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and nonexistence pieces of .