Homogeneous non-degenerate 3-(α,δ)-Sasaki manifolds and submersions over quaternionic Kähler spaces

We show that every 3-(α, δ) -Sasaki manifold of dimension 4n+ 3 admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature 16n(n+ 2)αδ. In the non-degenerate case we describe all homogeneous 3-(α, δ) -Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. If αδ> 0 , this yields a complete classification of homogeneous 3-(α, δ) -Sasaki manifolds. For αδ< 0 , we provide a general construction of homogeneous 3-(α, δ) -Sasaki manifolds fibering over non-symmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being 19.