@BenMcKay: Yes, it's easy to see, that in local coordinates they are given by a PDE. The problem is, that this depends on the chart. So the usual definition of a PDE (on a fibre bundle, here $M \times M \to M$), is that it is a closed submanifold of a convenient jet bundle. That's what I'm asking for. Maybe I should clarify this in my question.

@Sebastian: How can a curve in $\mathcal G$ be differentiable? As $\mathcal G$ is merely a Lie pseudogroup, it does not possess a differentiable structure. Thanks for helping!

Hi @Sebastian. I know how to calculate the commutator pointwise in terms of local flows. But I need the local flow of the commutator vector field to be contained in $\mathcal{G}$. I do not see why this is true, although I guess it is.