Flow of the sum of vector fields

In the case where $[V,W]=0$, it is pretty easy to show that

In the general non-commuting case, the flow ${\varphi }_{V+W}^t$ equals to first order both ${\varphi }_V^t\circ {\varphi }_W^t$ and ${\varphi }_W^t\circ {\varphi }_V^t$. Morally, the second order approximation should be 'halfway between' the two aforementioned flows. Since ${\varphi }_V^t\circ {\varphi }_W^t\circ {\varphi }_V^{-t}\circ {\varphi }_W^{-t}$ is approximated by ${\varphi }_{[V,W]}^{t^2}$, we expect to have

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.