  Using the relativistic impulse approximation with the Love-Franey \textsl{NN}
scattering amplitude developed by Murdock and Horowitz, we investigate the
low-energy (100 MeV$\leq E_{\mathrm{kin}}\leq 400$ MeV) behavior of the nucleon
Dirac optical potential, the Schr\"{o}dinger-equivalent potential, and the
nuclear symmetry potential in isospin asymmetric nuclear matter. We find that
the nuclear symmetry potential at fixed baryon density decreases with
increasing nucleon energy. In particular, the nuclear symmetry potential at
saturation density changes from positive to negative values at nucleon kinetic
energy of about 200 MeV. Furthermore,the obtained energy and density dependence
of the nuclear symmetry potential is consistent with those of the isospin- and
momentum-dependent MDI interaction with $x=0$, which has been found to describe
reasonably both the isospin diffusion data from heavy-ion collisions and the
empirical neutron-skin thickness of $^{208} $Pb.
