= Coordinate proof of the Frobenius theorem
For an <involutive distribution> of constant rank $r$, induct on $r$. Straighten a nonzero section by the <flow-box theorem>, and choose a frame $\partial_1,Y_2,\ldots,Y_r$ with no $\partial_1$ component in the remaining fields. Involutivity implies $\partial_1Y=A Y$ for their column $Y$. The invertible solution of $\partial_1B=-BA$, initialized by $B=I$ on $x_1=0$, makes $Z=BY$ independent of $x_1$. Its restriction to the transverse slice is involutive of rank $r-1$, so induction provides coordinates spanning it. Extending these coordinates independently of $x_1$ gives $D=\operatorname{span}(\partial_1,\ldots,\partial_r)$. Conversely, fields tangent to an <integral manifold> have tangent brackets because they preserve the ideal of smooth functions vanishing on it.
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