We use the procedure of reduction of Courant algebroids introduced in [H. BURSZTYN, G. R. CAVALCANTI, and M. GUALTIERI, Reduction of Courant algebroids and generalized complex structures. Adv. Math.., 211, 2007] to reduce strong KT, hyper KT and generalized Kähler structures on Courant algebroids. This allows us to recover results from the literature as well as explain from a different angle some of the features observed therein. As an example, we prove that the moduli space of instantons of a bundle over an SKT/HKT/generalized Kähler manifold is endowed with the same type of structure as the original manifold.
"Reduction of metric structures on courant algebroids." J. Symplectic Geom. 4 (3) 317 - 343, September 2006.