Commutators and coordinate-induced basis (General Relativity)

If the vectors form a basis then it is fairly easy to use Poincaré's lemma to show it, namely if $e_1,...,e_n$ is a frame, and $theta^1,...,theta^n$ is its dual frame, and the direct frame satisfies the commutation relations $$ [e_a,e_b]=C^c_{ ab}e_c, $$ then it is easy to show that the dual frame satisfies the differential relations $$ mathrm dtheta ^a=-frac{1}{2}C^a_{ bc}theta^bwedgetheta^c=-sum_{b<c}C^a_{ bc}theta^bwedge theta^c. $$

If the frame commutes, i.e. $C^c_{ ab}=0$, then we have $$ mathrm dtheta^a=0, $$ thus by Poincaré's lemma, each point in the frame's domain of definition has an open neighborhood $mathscr U$ on which functions $u^1,...,u^n$ exist such that $$ theta^a=mathrm du^a . $$

From this point, it is rather easy to see that the $u^1,...,u^n$ form a local coordinate system and that $$ e_a=frac{partial}{partial u^a}. $$

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If the $e_1,...,e_n$ do not form a frame (but are linearly independent, its just there are not enough of them to be a frame), the proof is more difficult. I am going to skip the details but this can be found in for example Lee's Introduction to smooth manifolds (called "canonical form for commuting vector fields" there).

The proof itself is based on the flowout/flow box/straightening theorem. Basically, we know that if $X$ is a vector field that does not vanish in the neighborhood of a point $x$, then there exists a coordinate system $x^1,...,x^n$ such that $$ X=frac{partial}{partial x^1}. $$

The sketch of the proof is that we can take a hypersurface $Sigma$ to which $X$ is transversal (not tangential), and we may put a coordinate system $x^2,...,x^n$ on the hypersurface, then use the flow $phi^X(t,x)$ of the vector field to construct a local foliation of hypersurfaces for which $t=x^1$ labels the individual hypersurfaces and $x^2,...,x^n$ labels a point on the $x^1$ hypersurface. Points with different $x^1$ but same $x^2,...,x^n$ lie on the same integral curve of $X$.

If one has $e_1,...,e_m$ commuting vector fields in the $n$ dimensional manifold $M$, then the basic idea of the proof is quite similar, take an $n-m$ dimensional submanifold to which the $e_1,...,e_m$ are all transversal, put a coordinate system $x^{m+1},...,x^n$ on the submanifold, define $$ phi_{x^1,...,x^m}(x)=phi^{e_1}_{x^1}circ...circphi^{e_m}_{x^m}(x), $$ where $phi^{e_i}_{x^i}$ is the flow of $e_i$ for time $x^i$.

The fact that the flows of the individual vector fields commute means that $$ phi_{x^1,...x^i,...,x^j,...,x^m}=phi_{x^1,...,x^j,...,x^i,...,x^m}, $$ which is crucial in proving that we actually have $$ e_i=frac{partial}{partial x^i} $$ once the construction is finished.