Flow of sum of non-commuting vector fields

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.