  The semileptonic decays of meson $B_c$ to a P-wave charmonium state
$\chi_c(^3P_J)$ or $h_c(^1P_1)$ are computed. The results show that the decays
are sizable so they are accessible in Tevatron and in LHC, especially, with the
detectors LHCB and BTeV in the foreseeable future, and of them, the one to the
$^1P_1$ charmonium state potentially offers us a novel window to see the
unconfirmed $h_c$ particle. In addition, it is pointed out that since the two
charmonium radiative decays $\chi_c(^3P_{1,2}) \to J/\psi+\gamma$ have sizable
branching ratios, the cascade decays of the concerned decays and the charmonium
radiative decays may affect the result of the observing the $B_c$ meson through
the semileptonic decays $B_{c}\to {J/\psi}+{l}+\nu_{l}$ substantially.
