Reducible Holonomy in Closed Torsion Geometries

The purpose of this note is to show that a connection with closed skew-symmetric torsion and reducible holonomy admits a locally defined Riemannian submersion together with a projected geometry on the base. We reframe known submersion results for non-Kähler Bismut-Hermite-Einstein manifolds and sHKT structures in this context. For homogeneous SKT structures on semi-simple Lie groups, we obtain the holonomy decomposition leading to holomorphic submersions over generalized flag manifolds.