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# Citeria for Involutive Subbundles

Preliminaries: Let $M$ be a smooth manifold with tangent bundle $TM$. A vector subbundle $VM$ of $TM$ is called involutive if the section space $\Gamma(VM)$ of $VM$ is closed under the Lie bracket of $\Gamma(TM)$ or in other words if $[X,Y] \in \Gamma(VM)$ for all $X,Y \in \Gamma(VM)$.

On the other side the Lie bracket of two vector field can be expressed entirely by the flow transformations of the fields, that is we have:

$$[X,Y] = \frac{1}{2}\frac{\partial^2}{\partial_t^2} |_{t=0}(Fl^Y _{-t}\circ Fl^X _{-t}\circ Fl^Y _{t}\circ Fl^X _{t})$$

where $Fl^X$ and $Fl^Y$ are the flow transformations of $X$ and $Y$ respectively.

Now the question is, can (and if yes, how) we decide whether or not o subbundle of $TM$ is involutive entirely in terms of flow transformations?

If somehow we want to proof that the bracket is closed on a subbundle but we know very little about it but instead know much about the associated flow transformations.