  Originally proposed as an $O(d,d)$-invariant formulation of classical closed
string theory, double field theory (DFT) offers a rich source of mathematical
structures. Most prominently, its gauge algebra is determined by the so-called
C-bracket, a generalization of the Courant bracket of generalized geometry, in
the sense that it reduces to the latter by restricting the theory to solutions
of a "strong constraint". Recently, infinitesimal deformations of these
structures in the string sigma model coupling $\alpha'$ were found. In this
short contribution, we review constructing the Drinfel'd double of a Lie
bialgebroid and offer how this can be applied to reproduce the C-bracket of DFT
in terms of Poisson brackets. As a consequence, we are able to explain the
$\alpha'$-deformations via a graded version of the Moyal-Weyl product in a
class of examples. We conclude with comments on the relation between $B$- and
$\beta$-transformations in generalized geometry and the Atiyah algebra on the
Drinfel'd double.
