Abstract
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.