  We construct a Lorentz-covariant connection in the context of first order
canonical gravity with non-vanishing Barbero-Immirzi parameter. To do so, we
start with the phase space formulation derived from the canonical analysis of
the Holst action in which the second class constraints have been solved
explicitly. This allows us to avoid the use of Dirac brackets. In this context,
we show that there is a "unique" Lorentz-covariant connection which is
commutative in the sense of the Poisson bracket, and which furthermore agrees
with the connection found by Alexandrov using the Dirac bracket. This result
opens a new way toward the understanding of Lorentz-covariant loop quantum
gravity.
