Reality conditions for Ashtekar gravity from Lorentz-covariant formulation

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.