  The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the
gauge symmetry algebra generated by the C-bracket in double field theory (DFT).
We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd
double of Lie algebroids. Based on a geometric realization of doubled
space-time as a para-Hermitian manifold, we examine exterior algebras and a
para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'd
double behind the DFT gauge symmetry. Similar to the Courant algebroid in the
generalized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a para-Hermitian
manifold play Dirac-like structures in the Vaisman algebroid. We find that an
algebraic origin of the strong constraint in DFT is traced back to the
compatibility condition needed for $(L,\tilde{L})$ be a Lie bialgebroid. The
analysis provides a foundation toward the "coquecigrue problem" for the gauge
symmetry in DFT.
