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Based on preliminary analysis of the Hamiltonian formulation of the firstorder Einstein-Cartan action (arXiv:0902.0856 [gr-qc] and arXiv:0907.1553[gr-qc]) we derive the Darboux coordinates, which are a unique and uniformchange of variables preserving equivalence with the original action in allspacetime dimensions higher than two. Considerable simplification of theHamiltonian formulation using the Darboux coordinates, compared with directanalysis, is explicitly demonstrated. Even an incomplete Hamiltonian analysisin combination with known symmetries of the Einstein-Cartan action and theequivalence of Hamiltonian and Lagrangian formulations allows us tounambiguously conclude that the \textit{unique} \textit{gauge} invariancesgenerated by the first class constraints of the Einstein-Cartan action and thecorresponding Hamiltonian are \textit{translation and rotation in the tangentspace}. Diffeomorphism invariance, though a manifest invariance of the action,is not generated by the first class constraints of the theory.

Darboux coordinates for the Hamiltonian of first order Einstein-Cartan gravity