Flow of sum of non-commuting vector fields Let $V,W\in\Gamma(M)$ be any two vector fields. Is there any "nice" expression for the flow of $V+W$ in terms of the flow of $V$ and the flow of $W$? It would be sufficient for me to have some sort of expansion in $\epsilon$ for the flow of $V+\epsilon W$, at least the first few terms (or even just at the order of $\epsilon$).
In the case where $[V,W]=0$, it is pretty easy to show that
$$\varphi_{V+W}^t = \varphi_V^t\circ\varphi_W^t,$$
but the non commuting case is of greater interest to me.

In the general non-commuting case, the flow $\phi^t_{V+W}$ equals to first order both $\phi^t_V \circ \phi^t_W$ and  $\phi^t_W \circ \phi^t_V$. Morally, the second order approximation should be 'halfway between' the two aforementioned flows. Since $\phi^{t}_V \circ \phi^{t}_W \circ \phi^{-t}_V \circ \phi^{-t}_W$ is approximated by $\phi^{t^2}_{[V,W]}$, we expect to have
$$ \phi^t_{V+W}(x)= \left(\phi^{t^2}_{\frac{1}{2}[V,W]} \circ \phi^t_W \circ \phi^t_V\right)(x) \, .$$
It happens to be the first few terms of the Zassenhaus formula (in reverse order) for the exponential map ; Notice that we can interpret a vector field on a manifold $M$ as an element of the Lie algebra of the infinite dimensional Lie group $\mathrm{Diff}(M)$, so that taking the exponential map $\mathfrak{diff}(M) \to \mathrm{Diff}(M)$ corresponds to integrating vector fields.
