  Using dispersive methods, we study the $B \to \gamma^*$ form factors
underlying the decay $B^- \to \ell^- \bar{\nu}_\ell \ell^{\prime-}
\ell^{\prime+}$. We discuss the ambiguity that arises from a separation of the
full $B^- \to \ell^- \bar{\nu}_\ell \ell^{\prime-} \ell^{\prime+}$ amplitude
into a hadronic tensor and a final-state-radiation piece, including effects
from nonvanishing lepton masses. For the eligibility of a dispersive treatment,
we propose a decomposition of the hadronic part that leads to four form factors
that are free of kinematic singularities. By establishing a set of dispersion
relations, we then relate the $B \to \gamma^*$ form factors to the well-known
$B \to V$, $V=\omega(782),\rho(770)$, analogues. Using the combination of a
series expansion in a conformal variable and a vector-meson-dominance ansatz to
parameterize the $B \to \gamma^*$ form factors, we infer the values of the
associated unknown parameters from the available input on $B \to V$. The
phenomenological application of our formalism includes the determination of the
branching ratios and forward-backward asymmetries of the process $B^- \to
\ell^- \bar{\nu}_\ell \ell^{\prime-} \ell^{\prime+}$ for the two lightest
leptons.
