Canonical Vielbeins for General Relativity: D + 1 Decomposition and Constraint Analysis

We provide a self-contained derivation of the Hamiltonian formulation of General Relativity in vielbein variables in $d=D+1$ dimensions. Starting from the Einstein–Hilbert action in a standard metric $D+1$ decomposition, we derive Lorentz- and $\mathrm{S}\mathrm{O}(D)$-covariant phase-space actions, identify the primary Lorentz constraints, and obtain the Hamiltonian and momentum constraints. We compute the resulting first-class constraint algebra, relate the vielbein and metric phase-space formulations, and discuss the rotation/boost decomposition. In particular, we construct the boost generator in the $\mathrm{S}\mathrm{O}(D)$-covariant formulation, thereby recovering full local Lorentz symmetry.