A general class of multi-parametric families of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered. A generalization of these commutation relations in three generators is also considered, modifying Lie algebra type commutation relations, typical for usual differential or difference operators, to relations satisfied by more general twisted difference operators associated with general twisting maps. General reordering and nested commutator formulas for arbitrary elements in these algebras are presented, and some special cases are considered, generalizing some well-known results in mathematics and physics. © Springer Nature Switzerland AG 2020.