  We consider the piecewise flat spacetime and a simplicial analog of the
Palatini form of the general relativity (GR) action where the discrete
Christoffel symbols are given on the tetrahedra as variables that are
independent of the metric. Excluding these variables classically gives exactly
the Regge action.
  This paper continues our previous work. Now we include the parity violation
term and the analogue of the Barbero-Immirzi parameter introduced in the
orthogonal connection form of GR. We consider the path integral and the
functional integration over connection. The result of the latter (for certain
limiting cases of some parameters) is compared with the earlier found result of
the functional integration over connection for the analogous {\it orthogonal}
connection representation of Regge action.
  These results, mainly as some measures on the lengths/areas, are discussed
for the possibility of the diagram technique where the perturbative diagrams
for the Regge action calculated using the measure obtained are finite. This
finiteness is due to these measures providing elementary lengths being mostly
bounded and separated from zero, just as finiteness of a theory on a lattice
with an analogous probability distribution of spacings.
