  In this note, we review the canonical analysis of the Holst action in the
time gauge, with a special emphasis on the Hamiltonian equations of motion and
the fixation of the Lagrange multipliers. This enables us to identify at the
Hamiltonian level the various components of the covariant torsion tensor, which
have to be vanishing in order for the classical theory not to depend upon the
Barbero-Immirzi parameter. We also introduce a formulation of three-dimensional
gravity with an explicit phase space dependency on the Barbero-Immirzi
parameter as a potential way to investigate its fate and relevance in the
quantum theory.
