  We complement and offer a new perspective of the proof of a Motohashi-type
formula relating the fourth moment of $L$-functions for $\mathrm{GL}_1$ with
the third moment of $L$-functions for $\mathrm{GL}_2$ over number fields,
studied earlier by Michel-Venkatesh and Nelson. Our main tool is a new type of
pre-trace formula with test functions on $\mathrm{M}_2(\mathbb{A})$ instead of
$\mathrm{GL}_2(\mathbb{A})$, on whose spectral side the matrix coefficients are
replaced by the standard Godement-Jacquet zeta integrals. This is also a
generalization of Bruggeman-Motohashi's other proof of Motohashi's formula. We
give a variation of our method in the case of division quaternion algebras
instead of $\mathrm{M}_2$, yielding a new spectral reciprocity, for which we
are not sure if it is within the period formalism given by Michel-Venkatesh. We
also indicate a further possible generalization, which seems to be beyond what
the period method can offer.
