Let \(M=G/K\) be a compact homogeneous space and assume that G and K have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that \(b_3(M)=s-1\) if G has s simple factors, so called aligned, leads to a relatively manageable algebraic structure on the isotropy representation, paving the way to the computation of Ricci curvature formulas for a large class of G-invariant metrics. As an application, we study the existence and classification of Einstein metrics on aligned homogeneous spaces.
