Metric algebroid and Dirac generating operator in Double Field Theory

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.