  A sterile neutrino with a mass of a few keV can play the role of a warm dark
matter(DM). This can be realized in seesaw models with 3 left- and 3
right-handed neutrinos. It is possible to identify the keV neutrino to be one
of the right-handed neutrinos leaving the other two to be much more heavier,
the $\nu$SM model. We show that with this realization of keV neutrino DM, the
model has an approximate Friedberg-Lee symmetry providing a natural explanation
for the lightness of the right-handed neutrino. We also find that in this model
the mixing parameters couple light and heavy neutrinos are strongly correlated,
and can be large enough to have testable effects at the LHC for the two heavy
right-handed neutrinos to be in the hundred-GeV range.
