In the case where $[V,W]=0$, it is pretty easy to show that
$$ \varphi_{V+W}^t = \varphi_V^t\circ\varphi_W^t. $$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.
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Author of the notes: Antonio J. Pan-Collantes
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