  The Cassini state equilibrium associated with the precession of the Moon
predicts that the mantle, fluid core and solid inner core precess at different
angles. We present estimates of the dissipation from viscous friction
associated with the differential precession at the core-mantle boundary (CMB),
$Q_{cmb}$, and at the inner core boundary (ICB), $Q_{icb}$, as a function of
the evolving lunar orbit. We focus on the latter and show that, provided the
inner core was larger than 100 km, $Q_{icb}$ may have been as high as
$10^{10}-10^{11}$ W for most of the lunar history for a broad range of core
density models. This is larger than the power required to maintain the fluid
core in an adiabatic state, therefore the heat released by the differential
precession at the ICB can drive a past lunar dynamo by thermal convection. This
dynamo can outlive the dynamo from precession at the CMB and may have shutoff
only relatively recently. Estimates of the magnetic field strength at the lunar
surface are of the order of a few $\mu$T, compatible with the lunar
paleomagnetic intensities recorded after 3 Ga. We further show that it is
possible that a transition of the Cassini state associated with the inner core
may have occurred as a result of the evolution of the lunar orbit. The heat
flux associated with $Q_{icb}$ can be of the order of a few mW m$^{-2}$, which
should slow down inner core growth and be included in thermal evolution models
of the lunar core.
