  Based on a general form of the effective vertex functions for the decays of
P-wave charmonia $\chicj$, angular distribution formulas for the subsequent
decays $\chicj\rightarrow \jpsi \gamma$ and $\jpsi \to \mu^+\mu^-$ are derived.
The formulas are the same as those obtained in a different approach in the
literature. Our formulas are expressed in a more general form, including parity
violation effects and the full angular dependence of $\jpsi$ and muon in the
cascade decay $\chicj\to\jpsi\gamma\to\mu^+\mu^-\gamma$. The $\chicj$
polarization observables are expressed in terms of rational functions of the
spin density matrix elements of $\chicj$ production. Generalized
rotation-invariant relations for arbitrary integer-spin particles are also
derived and their expressions in terms of observable angular distribution
parameters are given in the $\chi_{c1}$ and $\chi_{c2}$. To complement our
previous direct-$\jpsi$ polarization result, we also discuss the impact on the
observable prompt-$\jpsi$ polarization. As an illustrative application of our
angular distribution formulas, we present the angular distributions in terms of
the tree-level spin density matrix elements of $\chi_{c1}$ and $\chi_{c2}$
production in several different frames at the Large Hadron Collider. Moreover,
a reweighting method is also proposed to determine the entire set of the
production spin density matrix elements of the $\chi_{c2}$, some of which
disappear or are suppressed for vanishing higher-order multipole effects making
the complete extraction difficult experimentally.
