  Effects are investigated of CP symmetry violation in the decay of a scalar
particle X (the Higgs boson) into two off-mass-shell Z bosons both decaying
into a fermion-antifermion pair, $X \to Z_1^* \, Z_2^* \to f_1 \bar{f_1} \, f_2
\bar{f_2}$. The most general form of the amplitude of the transition $X \to
Z_1^* Z_2^*$, wherein the boson X may not have definite CP parity, is
considered. Limits of applicability of the narrow-Z-width approximation used
when obtaining differential widths of the decay under consideration are
determined. Various observables connected with the structure of the amplitude
of the decay $X \to Z_1^* Z_2^*$ are studied. These observables are analyzed in
the Standard Model, as well as in models conceding indefinite CP parity of the
Higgs boson. An experimental measurement at the LHC of angular and invariant
mass distributions of the decay $X \to Z_1^* Z_2^* \to f_1 \bar{f_1} f_2
\bar{f_2}$ can give information about the CP properties of the Higgs boson and
its interaction with the Z boson.
