Evaluation of the Lorentz group Lie algebra map using the Baker-Cambell-Hausdorff formula

This paper presents a formulation of a numerical integrator for charged particles. The original idea of the integrator scheme was given several years ago. The main topics of this paper is the reformulation of the integrator as the Lorentz group Lie algebra map and modifications for practical use. Two numerical simulation examples, trapped electron motion in a cusp magnetic field and electron pendulum motion in the free electron laser (FEL), show the validity of this new formulation.