The gauge group in the real triad formulation of general relativity

We construct explicitly generators of projectable four-dimensional diffeomorphisms and triad rotation gauge symmetries in a model of vacuum gravity where the fundamental dynamical variables in a Palatini formulation are taken to be a lapse, shift, densitized triad, extrinsic curvature, and the time-like components of the Ricci rotation coefficient. Time-foliation-altering diffeomorphisms are not by themselves projectable under the Legendre transformations. They must be accompanied by a metric- and triad-dependent triad rotation. The phase space on which these generators act includes all of the gauge variables of the model.