be formalized in Section 4. The crucial difference is dimensionality of the (pseudo-)group. In practice, almost always we must content ourselves with approximate solutions —Lie–Scheffers systems are not solvable by quadratures in general— and actually those are our main concern.

We reformulate  within the framework of Hopf algebra and Rota–Baxter operator theory —the latter has become popular recently in relation with the Connes–Kreimer paradigm for renormalization theory in perturbative quantum field theory. The convenience of such algebraic approach stems already from that, unless $G$ is a matrix group, $\dot g$ and $g^{-1}$ cannot be multiplied, and then equations  and  have no meaning, strictu sensu. Hopf algebras generalize both Lie groups and Lie algebras, so the problem does not present itself in a Hopf algebra formalism. Another advantage is that Hopf algebra and Rota–Baxter theory allow for efficient and meaningful comparisons among the different techniques for solving , proposed over the years. The main aim of this paper is to show these and other benefits of our algebraic viewpoint. They have been patent for a while to people working on control theory, but mostly ignored by the wider community of mathematicians and mathematical physicists.

We presume the readers acquainted with standard 