  From the Holst action in terms of complex valued Ashtekar variables
additional reality conditions mimicking the linear simplicity constraints of
spin foam gravity are found. In quantum theory with the results of You and
Rovelli we are able to implement these constraints weakly, that is in the sense
of Gupta and Bleuler. The resulting kinematical Hilbert space matches the
original one of loop quantum gravity, that is for real valued Ashtekar
connection. Our result perfectly fit with recent developments of Rovelli and
Speziale concerning Lorentz covariance within spin-form gravity.
