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I know that the following statement is true, but I would like to find a reference for it so I don't have to write the proof. Do you guys have a reference?

Let $\Omega$ and $\Omega'$ be smooth manifolds and let $\mathcal V \subset \mathbb C T \Omega$ be an involutive fiber bundle. Let $f : \Omega \to \Omega'$ be a submersion and suppose that for all smooth sections $X$ of $\ker f_*$ and all smooth sections $Y$ of $\mathcal V$, it holds that $[X, Y]$ is a smooth section of $\mathcal V$, then $f_*(V)$ is an involutive subbundle of $\mathbb C T\Omega'$.

Any help would be greatly appreciated.

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    $\begingroup$ This follows from the fact that $f_*[X,Y] = [f_*X,f_*Y]$, which is stated and proved in many textbooks on manifolds. $\endgroup$
    – Deane Yang
    Commented Nov 2, 2018 at 1:45
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    $\begingroup$ @DeaneYang, I don't see the point of you argument. Given two vector fields $X,Y$ you cannot assure that $f_* X$ and $f_* Y$ are well defined vector fields. I think this has to be proven by means of 1-forms... $\endgroup$
    – A. J. Pan-Collantes
    Commented Jul 24, 2022 at 8:06
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    $\begingroup$ @AntonioJPan, thanks for pointing this out. My comment does appear to be pointless. $\endgroup$
    – Deane Yang
    Commented Jul 24, 2022 at 15:42

1 Answer 1

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This is proved in Problem 2.57 of ``Analysis and algebra on differentiable manifolds: a workbook for students and teachers (2012)''.

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