Computing the Baker–Campbell–Hausdorff series and the Zassenhaus product

The Baker–Campbell–Hausdorff (BCH) series and the Zassenhaus product are of fundamental importance for the theory of Lie groups and their applications in physics and physical chemistry. Standard methods for the explicit construction of the BCH and Zassenhaus terms yield polynomial representations, which must be translated into the usually required commutator representation. We prove that a new translation proposed recently yields a correct representation of the BCH and Zassenhaus terms. This representation entails fewer terms than the well-known Dynkin–Specht–Wever representation, which is of relevance for practical applications. Furthermore, various methods for the computation of the BCH and Zassenhaus terms are compared, and a new efficient approach for the calculation of the Zassenhaus terms is proposed. Mathematica implementations for the most efficient algorithms are provided together with comparisons of efficiency.