  We show the equivalence of the Lorentz-covariant canonical formulation
considered for the Immirzi parameter $\beta=i$ to the selfdual Ashtekar
gravity. We also propose to deal with the reality conditions in terms of Dirac
brackets derived from the covariant formulation and defined on an extended
phase space which involves, besides the selfdual variables, also their
anti-selfdual counterparts.
