$[V,fW] = f [V,W] + V(f) W $ Lie product Some notation: Let $M$ be a smooth manifold and denote derivatives by $d$. For a vectorfield $V$ and $f \in C^\infty(M)$ we write $V(f)(p) = d_pf(v_p)$ where $v_p = V(p)$. Further $[V,W]$ is a vectorfield characterized by 
$$
[V,W](f) =V(f)W(f) - W(f)V(f).
$$
 Why does $[V,fW] = f [V,W] + V(f)W $ hold ? My computation yields for $f,g \in C^\infty(M)$ and $p \in M$  that 
\begin{align*}
[V,fW](g)(p)&  = V(g)(p)(fW)(g)(p) - (fW)(g)(p)V(g)(p) \\
& = d_pg(v_p)d_pg(f(p)w_p) - d_pg(f(p)w_p)d_pg(v_p) \\
& = f(p) (  d_pg(v_p)d_pg(w_p) - d_pg(w_p)d_pg(v_p)  ) \\
&= f(p)\cdot(V(g)(p)W(g)(p)-W(g)(p)V(g)(p))
\end{align*} Thus $[V,fW] = f [V,W]$.
What is wrong here ?

Your definition of the vector field $[V, W]$ is wrong (currently $[V, W]f = 0$ for any $f$). It should be $[V, W]f = V(Wf) - W(Vf)$.
