  We work on a spacetime manifold foliated by timelike leaves. In this setting,
we explore the solution of the second-class constraints arising during the
canonical analysis of the Holst action with a cosmological constant. The
solution is given in a manifestly Lorentz-covariant fashion, and the resulting
canonical formulation is expressed using several sets of real variables that
are related to one another by canonical transformations. By applying a gauge
fixing to this formulation, we obtain a description of gravity as an $SU(1,1)$
gauge theory that resembles the Ashtekar-Barbero formulation.
