We generalize an old result due to Lowenthal [1] and a more recent one due to Hamada [2] on the order of finite generation of the rotation group SO ( 3 ) both for fixed and arbitrary compound transformations. Unlike the above cited authors, we consider decompositions into factors with more than two invariant axes and provide rather intuitive geometric proofs. Thus, we derive a simple estimate for the number of factors in a decomposition and discuss possible means of optimization as well as particular examples of potential interest for the applications.

The paper provides a study of the commutative algebras generated by iteration of the cross products in $\mathbb{C}^3$. Focusing on particular real forms we also consider the analytical properties of the corresponding rings of functions and relate them to different physical problems. Familiar results from the theory of holomorphic and bi-holomorphic functions appear naturally in this context, but new types of hypercomplex calculi emerge as well. The parallel transport along smooth curves in $\mathbb{E}^3$ and the associated Maurer-Cartan form are also studied with examples from kinematics and electrodynamics. Finally, the dual extension is discussed in the context of screw calculus and Galilean mechanics; a similar construction is studied also in the multi-dimensional real and complex cases.