  We present the algebraic foundations of the symmetric Zassenhaus algorithm
and some of its variants. These algorithms have proven effective in devising
higher-order methods for solving the time-dependent Schr\"{o}dinger equation in
the semiclassical regime. We find that the favourable properties of these
methods derive directly from the structural properties of a Z2-graded Lie
algebra. Commutators in this Lie algebra can be simplified explicitly, leading
to commutator-free methods. Their other structural properties are crucial in
proving unitarity, stability, convergence, error bounds and quadratic costs of
Zassenhaus based methods. These algebraic structures have also found
applications in Magnus expansion based methods for time-varying potentials
where they allow significantly milder constraints for convergence and lead to
highly effective schemes. The algebraic foundations laid out in this work pave
the way for extending higher-order Zassenhaus and Magnus schemes to other
equations of quantum mechanics.
