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 ${\partial }_1,Y_2,\dots ,Y_r$ with no ${\partial }_1$ component in the remaining fields. Involutivity implies ${\partial }_{1}Y=AY$ for their column $Y$. The invertible solution of ${\partial }_{1}B=-BA$, 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 $D=\mathrm{span}({\partial }_1,\dots ,{\partial }_r)$. Conversely, fields tangent to an integral manifold have tangent brackets because they preserve the ideal of smooth functions vanishing on it.