  We have investigated the spin dynamics of a high-mobility two-dimensional
electron system (2DES) in a GaAs--Al$_{0.3}$Ga$_{0.7}$As single quantum well by
time-resolved Faraday rotation (TRFR) in dependence on the initial degree of
spin polarization, $P$, of the 2DES. From $P\sim 0$ to $P\sim 30$ %, we observe
an increase of the spin dephasing time, $T_2^\ast$, by an order of magnitude,
from about 20 ps to 200 ps, in good agreement with theoretical predictions by
Weng and Wu [Phys. Rev. B {\bf 68}, 075312 (2003)]. Furthermore, by applying an
external magnetic field in the Voigt configuration, also the electron $g$
factor is found to decrease for increasing $P$. Fully microscopic calculations,
by numerically solving the kinetic spin Bloch equations considering the
D'yakonov-Perel' and the Bir-Aronov-Pikus mechanisms, reproduce the most
salient features of the experiments, {\em i.e}., a dramatic decrease of spin
dephasing and a moderate decrease of the electron $g$ factor with increasing
$P$. We show that both results are determined dominantly by the Hartree-Fock
contribution of the Coulomb interaction.
