  Experimental measurements of decay branching fractions of semitaunic and
semimuonic $B_c$ into $J/\psi$ has challenged the lepton flavor universality in
standard model with about two standard deviations. In this paper, we first
investigate the unitary constraint on form factors of $B_c$ meson into $S$-wave
and $P$-wave charmonium. Such constraint leads to the exploration of the
$R_{J/\psi}$ and other ratios $R_{\eta_c}$, $R_{h_c}$, and $R_{\chi_{cJ}}$ in a
model-independent way. These results together with future experimental
measurements can be used to explore the lepton flavor universality in a more
systematic way. In addition, we point out that the helicity-dependent ratios
$R^L_{J/\psi}$ and $R^\perp_{J/\psi}$ can also provide complementary
information.
