  This work gives a Lie operator derivation of various Boris solvers via a
detailed study of trajectory errors in a constant magnetic field. These errors
in the gyrocenter location and the gyroradius are the foundational basis for
why Boris solvers existed, independent of any finite-difference schemes. This
work shows that there are two distinct ways of eliminating these errors so that
the trajectory of a charged particle in a constant magnetic field is exactly on
the cyclotron orbit. One way reproduces the known second-order symmetric Boris
solver. The other yields a previously unknown, but also on-orbit solver, not
derivable from finite-difference schemes. By revisiting some historical
calculations, it is found that many publications do not distinguish the poorly
behaved leap-frog Boris solver from the symmetric second-order Boris algorithm.
This symmetric second-order Boris solver's trajectory is much more accurate and
remains close to the exact orbit in a combined $nonuniform$ electric and
magnetic field at time-steps greater than the cyclotron period. Finally, this
operator formalism showed that Buneman's cycloid fitting scheme is
mathematically identical to Boris' on-orbit solver and that Boris' E-B
splitting is unnecessary.
