  The electrical resistivity of Fermi liquids (FLs) displays a quadratic
temperature ($T$) dependence because of electron-electron (e-e) scattering. For
such collisions to decay the charge current, there are two known mechanisms:
inter-band scattering (identified by Baber) and Umklapp events. However, dilute
metallic strontium titanate (STO) was found to display $T^2$ resistivity in
absence of either of these two mechanisms. The presence of soft phonons and
their possible role as scattering centers raised the suspicion that $T$-square
resistivity in STO is not due to e-e scattering. Here, we present the case of
Bi$_2$O$_2$Se, a layered semiconductor with hard phonons, which becomes a
dilute metal with a small single-component Fermi surface upon doping. It
displays $T$-square resistivity well below the degeneracy temperature where
neither Umklapp nor interband scattering is conceivable. We observe a universal
scaling between the prefactor of $T^2$ resistivity and the Fermi energy, which
is an extension of the Kadowaki-Woods plot to dilute metals. Our results imply
the absence of a satisfactory theoretical basis for the ubiquity of e-e driven
$T$-square resistivity in Fermi liquids.
