AbstractWe define the notions of$$B_{n}$$Bn-generalized pseudo-Hermitian and$$B_{n}$$Bn-generalized pseudo-Kähler structure on an odd exact Courant algebroidE. WhenEis in the standard form (or of type$$B_{n}$$Bn) we express these notions in terms of classical tensor fields on the base ofE. This is analogous to the bi-Hermitian viewpoint on generalized Kähler structures on exact Courant algebroids. We describe left-invariant$$B_{n}$$Bn-generalized pseudo-Kähler structures on Courant algebroids of type$$B_{n}$$Bnover Lie groups of dimension two, three and four.