  Polarized and unpolarized observables in the process of the triplet
photoproduction on a free electron, $\gamma +e\to e+e+e$, have been calculated
in the laboratory system. These observables are calculated in the approximation
when the Borsellino and $\gamma -e$ diagrams are taken into account. Using the
method of the invariant integration over the produced electron-positron pair
variables, the different distributions were obtained in the analytical form. We
obtain the double distribution over the $q^2$ (the square of the four-momentum
transfer to the recoil electron) and $Q^2$ (the created $e^+e^-$-pair invariant
mass squared) variables, and single distributions over $q^2$ or $Q^2$
variables. We consider the following cases: unpolarized particles, the photon
beam is linearly polarized and circularly polarized photon beam interacts with
a polarized electron target. The influence of the $\gamma -e$ terms on the
calculated observables has been investigated. The possibility to use this
process for the measurement of the circular polarization of the high-energy
photon beam is also considered. The numerical estimations of various
polarization observables have been done.
