First-order tetrad gravity in Dirac's Hamiltonian formalism

The Hamiltonian $\int d^{3}x(p\dot{q}-\mathcal{L})$ is constructed without adding further terms. The velocities ${\dot{e}}_{\mathrm{ia}}$ and ${\dot{\omega }}_{\mathrm{iab}}$ are eliminated while ${\dot{e}}_{0a}$ and ${\dot{\omega }}_{0ab}$ remain arbitrary. The second-class constraints reduce the theory to second-order tetrad gravity. The first-class constraints differ from those in second-order formalism, but satisfy the same gauge algebra provided one uses Dirac brackets.