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I am missing something but I can't see what, hopefully someone can help.

Suppose we have a connection $H$ in priciple $G$ bundle $G \rightarrow P \rightarrow M$. Than $TP=H \oplus V$. Sections of $H$ are locally spanned by lifts of vector fields on $M$. Lie bracket of lifts $\hat X, \hat Y $is a lift of a lie bracket $\hat{[X,Y]}$. In consequence one would obtain that $H$ is integrable since any two sections from $\Gamma(H)$ are written as cobination of lifts (locally) and their barcket is than combination of that lifts and their lie brackets which are in $H$.

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It is false that the bracket of lifts is the lift of brackets. let $M=\bf R^2$ and $G= \bf R$, so that $P= \bf R^3$ with coordinates $(x,y,z)$. Let $H= \ker \omega$, with $\omega = dz -xdy$. Let $X= \partial _x$, $Y=\partial _y$. Then $\hat X= \partial _x$, $\hat Y= x\partial _z+\partial _y$, and the bracket $[\hat X, \hat Y]=\partial _z$, whereas $[X,Y]=0$

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    Thanks very much. Actually a bracket of lifts is a lift (when lift means that the fields projects to a field over basis) yet in the previous post I (wrongly) thought about horizontal lifts and than it is folse, as I understood thanks to your example. – 2017-02-17