  We present a measurement of the ratio of partial branching fractions of the
semi-leptonic inclusive decays, $\overline{B} \to X_{u} \ell \overline{\nu}$ to
$\overline{B} \to X_{c} \ell \overline{\nu}$, where $\ell = (e, \mu)$, using
the full Belle sample of $772 \times 10^{6}$ $B \kern 0.18em\overline{\kern
-0.18em B}$ pairs collected at the $\Upsilon(4S)$ resonance. The ratio is
measured via a two-dimensional fit to the squared four-momentum transfer to the
lepton pair, and the charged lepton energy in the $B$ meson rest frame, where
the latter must be larger than $1$ Ge\kern -0.1em V, covering approximately
$86\%$ and $78\%$ of the $\overline{B} \to X_{u} \ell \overline{\nu}$ and
$\overline{B} \to X_{c} \ell \overline{\nu}$ phase space, respectively. We find
$\Delta \mathcal{B}(\overline{B} \to X_{u} \ell \overline{\nu})/ \Delta
\mathcal{B}(\overline{B} \to X_{c} \ell \overline{\nu}) = 0.0196(1 \pm
8.4\%_{\rm stat} \pm 7.9\%_{\rm syst})$ where the uncertainties are statistical
and systematic, respectively. In addition, we report the partial branching
fractions separately for charged and neutral $B$ meson decays, and for electron
and muon decay channels. We place a limit on isospin breaking in $\overline{B}
\to X_{u} \ell \overline{\nu}$ decays, and find no indication of lepton flavor
universality violation in either the charmed or charmless mode. Furthermore, we
unfold the $\overline{B} \to X_{u} \ell \overline{\nu}$ and $\overline{B} \to
X_{c} \ell \overline{\nu}$ yields and report the differential ratio in lepton
energy and four-momentum transfer squared.
