A note on the Baker--Campbell--Hausdorff series in terms of right-nested commutators

TL;DR

This paper presents compact expressions for the Baker--Campbell--Hausdorff series using right-nested commutators, reducing the number of terms compared to classical methods up to grade 10.

Contribution

It introduces a new approach to express the BCH series with fewer terms by leveraging identities among right-nested commutators.

Findings

Fewer terms in BCH series up to grade 10 compared to classical basis.

Explicit expressions involving independent commutators.

Complete set of identities among right-nested commutators derived.

Abstract

We get compact expressions for the Baker--Campbell--Hausdorff series $Z = \log(\e^X \, \e^Y)$ in terms of right-nested commutators. The reduction in the number of terms originates from two facts: (i) we use as a starting point an explicit expression directly involving independent commutators and (ii) we derive a complete set of identities arising among right-nested commutators. The procedure allows us to obtain the series with fewer terms than when expressed in the classical Hall basis at least up to terms of grade 10.