Having tried to solve exercise 4.4-7 to have another proof of Frobenius Theorem, I would ask you a question.
This is what I have understood: In Step 2 there is to prove, for any tangent subbundle $E$ on a manifold $M,$ that if the module $\Gamma(E)$ of its sections is a Lie subalgebra of $\Gamma(TM)$ then it is locally generated by $k$ independent commuting vector fields. In Step 1 there is to prove the integrability of $E$ when it is locally generated by $k$ independent commuting vector fields.
Now this is the question:
Because, in the step 1, the authors assume additionally that $\Gamma(E)$ has to be an abelian Lie subalgebra of $\Gamma(TM)$?
There is some point that I don't understand? or is it a lapsus calami?
Thank you very much for the attention.