Problem: Let $V, W$ be two complete vector fields, with $[V, W] = 0$. Denote $\Phi^V_t$ and $\Phi^W_t$ as the flows of the two vector fields respectively. Prove that $\Phi^{V+W}_t = \Phi^V_t \Phi^W_t$.
Attempt: this seems like a simple exercise of the relationship between vector fields and flows but I was not able to get the exact answer and use the fact that the two vector fields commute, so please point out mistake/missing steps in my solution please.
My calculations: $\partial_t|_{t=0}(\Phi^V_t \Phi^W_t)(x) = (\partial_t|_{t=0} \Phi^V(t,\Phi^W(t=0, x))) + (\partial_y|_{y=\Phi{^W(t=0,x) = x}} \Phi^V(t,y))(\partial_t|_{t=0} \Phi^W(t,x)) = V_{x} + (\partial_y|_{y=x} \Phi^V(t,y))W_x$
Any help is appreciated!
We can denote the flows by the exponential map: $\Phi_t^V=\exp(tV)$ and $\Phi_t^W=\exp(tW)$. By the Baker–Campbell–Hausdorff formula, if $[V,W]=0$, then $$\exp(V)\exp(W)=\exp(V+W).$$ This gives $\Phi_t^V\Phi_t^W=\Phi_t^{V+W}$.