Sorry for the delay; I’ve been swamped at my actual job the last couple days.
Today I want to show a certain converse: if we have vector fields on some open region that are linearly independent at some and which commute — for all and — then we can find some coordinate chart around so that the are the first coordinate vector fields. That is,
In particular, if is a vector field with then there is a coordinate chart around with .
If is any chart, then we can describe the th coordinate vector field by saying it’s the unique vector field on that is -related to the th partial derivative in . That is, we’re trying to prove that: on .
In fact, we can further simplify our claim by assuming that , , and — that the vector fields agree at the point . Indeed, if is any coordinate map taking to then we can define the vector fields on . These must have vanishing brackets because we can calculate:
What’s more, if is a local diffeomorphism of with , then is a coordinate map satisfying our assertion.
Now, let be the flow of , and let be a small enough neighborhood of that we can define by
The order that the flows come in here doesn’t matter, since we’re assuming that the — and thus their flows — commute. Anyway, given any smooth test function on we can check
That is, . To see this for any other , simply swap around the flows to bring to the front.
We can also check that
Thus is the identity transformation on . The inverse function theorem now tells us that there is a chart around with , which will then satisfy our assertions.