  We define the notions of B_n-generalized pseudo-Hermitian and B_n-generalized
pseudo-Kahler structures on an odd exact Courant algebroid E. When E is in the
standard form (or of type B_n) we express these notions in terms of classical
tensor fields on the base of E. This is analogous to the bi-Hermitian viewpoint
on generalized Kahler structures on exact Courant algebroids. We describe
left-invariant B_n-generalized pseudo-Kahler structures on Courant algebroids
of type B_n over Lie groups of dimension two, three and four.
