Let $\mathcal{G}$, $M$ and $B$ be tree smooth manifolds. And $s:\mathcal{G}\rightarrow M$ a surjective submersion and $\pi:M\rightarrow B$ a surjective submersion. Consider the vertical bundle given by $Ker(d\pi\circ ds)$. If $X$ is a vertical vector field and $Y$ is a projectable vector field how i can prove that $[X,Y]$ is a vertical vector field?
$[X,Y](f)$, where $f:\mathcal{G}\rightarrow\mathbf{R}$ is a smooth function constant on fiber of $\pi\circ s$, is given by $X(Y(f))-Y(X(f))$. Since $X$ is a vertical $Y(X(f))=0$ but what can i say about $X(Y(f))$? I think that $X(Y(f))=X(V(f\circ s))=X(\partial(f\circ s\circ \pi))=0$. Is it right? where $V$ and $\partial$ are projections of $Y$ I don't know if it is the right argumentation.
Thank for all help me.