  We continue to study doubled aspects of algebroid structures equipped with
the C-bracket in double field theory (DFT). We find that a family of
algebroids, the Vaisman (metric or pre-DFT), the pre- and the ante-Courant
algebroids are constructed by the analogue of the Drinfel'd double of Lie
algebroid pairs. We examine geometric implementations of these algebroids in
the para-Hermitian manifold, which is a realization of the doubled space-time
in DFT. We show that the strong constraint in DFT is necessary to realize the
doubled and non-trivial Poisson structures but can be relaxed for some
algebroids. The doubled structures of twisted brackets and those associated
with group manifolds are briefly discussed.
