Sasakian manifolds with purely transversal Bach tensor

We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature $≥2n(2n+1)$⁠, equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S³. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π₁(M).