Einstein metrics on aligned homogeneous spaces with two factors

Given two homogeneous spaces of the form G₁/K and G₂/K, where G₁ and G₂ are compact simple Lie groups, we study the existence problem for G₁xG₂-invariant Einstein metrics on the homogeneous space M=G₁xG₂/K. For the large subclass C 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 non-existence pieces of C.