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The properties of multicomponent fluids are governed by the interplay of phase behavior, fluid dynamics, and interfacial thermodynamics. A mixture formulation that leverages this interplay is an important aspect in many fabrication processes based on emulsion templating. The lattice Boltzmann method (LBM) has become a popular approach for simulating hydrodynamic effects in complex fluids and soft matter. Here we present an implementation of a ternary lattice Boltzmann model that allows to simulate a mixture of three immiscible fluids. We build on the LATBOLTZ extension of the open-source package LAMMPS and implement a ternary free energy model recently introduced by Semprebon et al. [Phys. Rev. E 93, 033305 (2016)]. We validate the static and dynamic properties by simulating liquid lenses, double emulsions, and ternary mixtures. From the simulations, we obtain the complete morphology diagram of the ternary mixture in composition space. We further discuss an application of the method to phase segregation of ternary films. The implementation of the ternary LBM in LAMMPS opens vast opportunities for mesoscale simulations of interfacial phenomena and non-equilibrium transport processes in multicomponent fluid mixtures.