  In this paper, we consider the elliptic relative equilibria of four-body
problem. Here we prove that the corresponding linearized Hamiltonian system at
such an elliptic relative equilibria of $4$-bodies splits into two independent
linear Hamiltonian systems, the first one is the linearized Hamiltonian system
of the Kepler $2$-body problem at Kepler elliptic orbit, and the other system
is the essential part of the linearized Hamiltonian system, which is given
implicitly. The reduction can be applied to the stability problem of such
elliptic relative equilibria of four-body problem.
