  In this paper, we consider the linear stability of the elliptic relative
equilibria of the restricted 4-body problems where the three primaries form a
Lagrangian triangle. By reduction, the linearized Poincar\'e map is decomposed
to the essential part, the Keplerian part and the elliptic Lagrangian part
where the last two parts have been studied in literature. The linear stability
of the essential part depends on the masses parameters $\alpha$, $\beta$ with
$\alpha\geq \beta >0$ and the eccentricity $e\in[0,1)$. Via $\om$-Maslov index
theory and linear differential operator theory, we obtain the full bifurcation
diagram of linearly stable and unstable regions with respect to $\alpha$,
$\beta$ and $e$. Especially, two linearly stable sub-regions are found.
