  We study the semileptonic decay of the $B_c (0^-)$ to charmonia through the
bottom-to-charm-quark electroweak current in the framework of basis light-front
quantization. Explicitly, we calculate the weak transition form factors for
processes of $B_c$ decaying into $\eta_c$ or $J/\psi$ based on the
corresponding initial and final valence light-front wave functions both
obtained from the basis light-front quantization. We also present the
corresponding differential decay width and branching ratios, as well as the
branching ratios for decays into the $\eta_c'$ and the $\psi'$. We observe
unphysical frame dependence of the calculated form factors, which is attributed
to only including the valence light-front wave functions. Based on the analysis
of current component, we propose a preferred set of frames to calculate these
form factors.
