Canonical form of commuting vector fields

Also known as Simultaneous Flow Box Theorem.
Theorem(Canonical Form for Commuting Vector Fields).
Let $M$ be a smooth $n$-manifold, and let $V_1,\dots ,V_k$ be smooth vector fields on an open subset of $M$ whose values are linearly independent at each point. Then the following are equivalent:
(a) There exist coordinates $(x^i)$ in a neighborhood of each point such that $V_i=\partial /\partial x^i$, $i=1,\dots ,k$.
(b) $[V_i,V_j]\equiv 0$ for all $i$ and $j$.
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See @lee2013smooth.
It is used to prove Frobenius theorem.