  The hot-electron effect in the spin relaxation of electrically injected
electrons in intrinsic Germanium is investigated by the kinetic spin Bloch
equations both analytically and numerically. It is shown that in the
weak-electric-field regime with $E\lesssim 0.5$~kV/cm, our calculations has
reasonable agreement with the recent transport experiment in the spin-injection
configuration [Phys. Rev. Lett. {\bf 111}, 257204 (2013)]. We reveal that the
spin relaxation is significantly enhanced at low temperature in the presence of
weak electric field $E\lesssim 50$~V/cm, which originates from the obvious
center-of-mass drift effect due to the weak electron-phonon interaction,
whereas the hot-electron effect is demonstrated to be less important. This can
explain the discrepancy between the experimental observation and the previous
theoretical calculation [Phys. Rev. B {\bf 86}, 085202 (2012)], which deviates
from the experimental results by about two orders of magnitude at low
temperature. It is further shown that in the strong-electric-field regime with
$0.5\lesssim E \lesssim 2$~kV/cm, the spin relaxation is enhanced due to the
hot-electron effect, whereas the drift effect is demonstrated to be marginal.
Finally, we find that when $1.4 \lesssim E\lesssim 2$~kV/cm which lies in the
strong-electric-field regime, a small fraction of electrons ($\lesssim 5\%$)
can be driven from the L to $\Gamma$ valley, and the spin relaxation rates are
the same for the $\Gamma$ and L valleys in the intrinsic sample without
impurity. With the negligible influence of the spin dynamics in the $\Gamma$
valley to the whole system, the spin dynamics in the L valley can be measured
from the $\Gamma$ valley by the standard direct optical transition method.
