  We study an analogue of the Drinfel'd double for algebroids associated with
the $O(D,D+n)$ gauged double field theory (DFT). We show that algebroids
defined by the twisted C-bracket in the gauged DFT are built out of a direct
sum of three (twisted) Lie algebroids. They exhibit a "tripled", which we call
the extended double, rather than the "doubled" structure appeared in (ungauged)
DFT. We find that the compatibilities of the extended doubled structure result
not only in the strong constraint but also the additional condition in the
gauged DFT. We establish a geometrical implementation of these structures in a
$(2D+n)$-dimensional product manifold and examine the relations to the
generalized geometry for heterotic string theories and non-Abelian gauge
symmetries in DFT.
