  We extend the techniques of double field theory to more general gravity
theories and U-duality symmetries, having in mind applications to the complete
D=11 supergravity. In this paper we work out a (3+3)-dimensional `U-duality
covariantization' of D=4 Einstein gravity, in which the Ehlers group SL(2,R) is
realized geometrically, acting in the 3 representation on half of the
coordinates. We include the full (2+1)-dimensional metric, while the `internal
vielbein' is a coset representative of SL(2,R)/SO(2) and transforms under gauge
transformations via generalized Lie derivatives. In addition, we introduce a
gauge connection of the `C-bracket', and a gauge connection of SL(2,R), albeit
subject to constraints. The action takes the form of (2+1)-dimensional gravity
coupled to a Chern-Simons-matter theory but encodes the complete D=4 Einstein
gravity. We comment on generalizations, such as an `$E_{8(8)}$
covariantization' of M-theory.
