This question is related to the following rather deep result: if $G$ is a finite dimensional Lie group, then $\pi_2(G)=0$.
Note that the
The analogous statement fails for Lie algebroids, as does Lie III. The paper of Tseng and Zhu shows that this is no accident.
Note that
Lie III may be interpreted as saying the following: the foliation of the space of $\frak{g}$-connections on the 1-simplex associated to infinitesimal gauge transformation has a Hausdorff leaf space.
A connection is a one-form $A\in\mathcal{A}=\Omega^1([0,1],\frak{g})$ with values in the Lie algebra $\frak{g}$. The infinitesimal gauge action is given by the formula
$\delta_XA=dX+[A,X]$,
where $X\in\Omega^0([0,1],\frak{g})$: these vectors span an integrable distribution in the tangent space of $\mathcal{A}$.
(This leaf space is then the simply connected Lie group associated to $G$.) I was taught this point of view by Raoul Bott.
