  We give a formulation of Double Field Theory (DFT) based on a metric
algebroid. We derive a covariant completion of the Bianchi identities, i.e. the
pre-Bianchi identity in torsion and an improved generalized curvature, and the
pre-Bianchi identity including the dilaton contribution. The derived bracket
formulation by the Dirac generating operator is applied to the metric
algebroid. We propose a generalized Lichnerowicz formula and show that it is
equivalent to the pre-Bianchi identities. The dilaton in this setting is
included as an ambiguity in the divergence. The projected generalized
Lichnerowicz formula gives a new formulation of the DFT action. The closure of
the generalized Lie derivative on the spin bundle yields the Bianchi identities
as a consistency condition. A relation to the generalized supergravity
equations (GSE) is discussed.
