  We perform the canonical analysis of the Holst action for general relativity
with a cosmological constant without introducing second-class constraints. Our
approach consists in identifying the dynamical and nondynamical parts of the
involved variables from the very outset. After integrating out the nondynamical
variables associated with the connection, we obtain the description of phase
space in terms of manifestly $SO(3,1)$ [or $SO(4)$, depending on the signature]
covariant canonical variables and first-class constraints only. We impose the
time gauge on them and show that the Ashtekar-Barbero formulation of general
relativity emerges. Later, we discuss a family of canonical transformations
that allows us to construct new $SO(3,1)$ [or $SO(4)$] covariant canonical
variables for the phase space of the theory and compare them with the ones
already reported in the literature, pointing out the presence of a set of
canonical variables not considered before. Finally, we resort to the time gauge
again and find that the theory, when written in terms of the new canonical
variables, either collapses to the $SO(3)$ ADM formalism or to the
Ashtekar-Barbero formalism with a rescaled Immirzi parameter.
