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

The symplectic reduction of the linearized Hamiltonian systems at elliptic relative equilibria of four-body problem