In this paper, we revise the concept of noncommutative vector fields
introduced previously in Ref. [1,2], extending the framework, adding new
results and clarifying the old ones. Using appropriate algebraic tools certain
shortcomings in the previous considerations are filled and made more precise.
We focus on the correspondence between so-called Cartan pairs and first-order
differentials. The case of free bimodules admitting more friendly “coordinate
description” and their braiding is considered in more detail. Bimodules of
right/left universal vector fields are explicitly constructed.
