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We demonstrate that from the first order formulation of the Einstein-Cartanaction it is possible to derive the basic differential identity that leads totranslational invariance of the action in the tangent space. Thetransformations of fields is written explicitly for both the first and secondorder formulations and the group properties of transformations are studied.This, combined with the preliminary results from the Hamiltonian formulation(arXiv:0907.1553 [gr-qc]), allows us to conclude that without any modification,the Einstein-Cartan action in any dimension higher than two possesses not onlyrotational invariance but also a form of \textit{translational invariance inthe tangent space}. We argue that \textit{not} only a complete Hamiltoniananalysis can unambiguously give an answer to the question of what a gaugesymmetry is, but also the pure Lagrangian methods allow us to find the samegauge symmetry from the \textit{basic} differential identities.

Translational invariance of the Einstein-Cartan action in any dimension