  The decays $B_c^+ \to B_{a} \bar \ell \nu_\ell$ and $B_c^+ \to B_{a}^{*}(\to
B_a \gamma) \bar \ell \nu_\ell$, with $a=s,d$ and $\ell=e,\mu$, are studied in
the Standard Model (SM) and in the extension based on the low-energy
Hamiltonian comprising the full set of dimension-$6$ semileptonic $c \to s,d$
operators with left-handed neutrinos. Tests of $\mu/e$ universality are
investigated using such modes. The heavy quark spin symmetry is applied to
relate the relevant hadronic matrix elements and to exploit lattice QCD results
on $B_c$ form factors. Optimized observables are selected, and the pattern of
their correlations is studied to identify the effects of the various operators
in the extended low-energy Hamiltonian.
