Double field theory (DFT) is an effective theory of string theory. It has a manifest symmetry of T-duality. The gauge symmetry in DFT is related to some kind of algebroid structures, and they have a doubled structure. In this talk, we focus on the gauge algebra of the $O(D, D + n)$ gauged DFT and discuss an extension of the doubled structure. The gauge algebra of the $O(D, D+n)$ gauged DFT has been described by the twisted C-bracket. This bracket is related to some algebroid structures. 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 ``triple'', which we call the extended double, rather than a ``double” structure. We also consider the geometrical realization of these structures in a $(2D+n)$-dimensional manifold.