Can anyone help me with a proof, I spent hours on it and nothing: I want to show $$\overline{[fX, Y]}=f\overline{[X, Y]},$$ where $X$ and $Y$ are smooth vector fields on a smooth manifold $M$, $f\in C^\infty(M)$ and $\overline{Z}$ represents the equivalence class which defines the quotient space $TM/F$ where $TM$ is the tangent bundle and $F$ is an involutive smooth distribution (in our case, for example, $Z\in \overline{[X, Y]}$ if and only if $Z-[X, Y]\in F$). Here $[, ]$ is the Lie bracket of smooth vector fields.
I guess this informations are the only needed for proving this fact.. I already proved the inclusion $f\overline{[X, Y]}\subset \overline{[fX, Y]}$, the problem is the converse..
This question is related to the Bott connection on the normal bundle.