The third Betti number of a compact homogeneous space M=G/K, where G has s simple factors and dim(K)>0, is always less or equal than s-1.  Equality holds if and only if certain s-vectors defined by the Killing constants are all collinear and in that case, M=G/K is called aligned.  In this talk, we will give formulas for the Ricci curvature of G-invariant metrics on aligned homogeneous spaces with s=2 and present two applications: 

1) New examples of generalized Einstein metrics (or Bismut Ricci flat generalized metrics (g,H), i.e., the fixed points of the generalized Ricci flow studied by Garcia-Fernandez and Streets), as a generalization of results by Podestà and Raffero. 

2) New examples of Einstein metrics in the largely unexplored case of compact homogeneous spaces G/K with G non-simple.