  In this note, using a representation theoretic method of Cogdell and
Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary
discrete group $\Gamma$ in $\mathrm{PGL}_2(\mathbb{C})$ that is cofinite but
not cocompact. An essential ingredient is a kernel formula, recently proved by
the author, on Bessel functions for $\mathrm{PGL}_2(\mathbb{C})$. This approach
avoids the difficult analysis in the existing method due to Bruggeman and
Motohashi.
