  Motivated by the recent experiments on the triangular lattice antiferromagnet
FeI$_2$, we consider the presence of the quadrupole moments and their
interactions between the spin-orbital-entangled local moments. In addition to
the anisotropic pairwise interaction between the local moments, the interaction
between the quadrupole moment arises from the interaction of the orbital
occupation configurations. This is argued primarily from the time reversal
symmetry and the different orbital configurations via their exchange paths. We
discuss the implication of these interactions and expect this result to be
complementary to the existing works in this system and other systems alike.
