Recall the following classical theorem of Cartan (!):


**Theorem (Lie III):** Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group.


Similarly, one can propose "Lie III" statements for Lie algebras over other fields, for super Lie algebras, for Lie algebroids, etc.


The proof I know of the classical Lie III is very difficult: it requires most of the structure theory of Lie algebras.


But why should it be difficult? For example, for finite-dimensional Lie algebra $\mathfrak g$ over $\mathbb R$, the Baker-Campbell-Hausdorff formula (the power series given by $B(x,y) = \log(\exp x \exp y)$ in noncommuting variables $x,y$; it can be written with only the Lie bracket, no multiplication) converges in an open neighborhood of the origin, and so defines a unital associative partial group operation on (an open neighborhood in) $\mathfrak g$. What happens if one were to try to simply glue together copies of this open neighborhood?


Alternately, are there natural variations of Lie III that are so badly false that any easy proof of Lie III is bound to fail?



Why is Lie's Third Theorem difficult?
That gluing together of group chunks, constructed from the BCH formula is precisely more or less what Serre does to prove the theorem (in the first proof he gives in) his book on Lie groups and Lie algebras. [Serre, Jean-Pierre. Lie algebras and Lie groups. 1964 lectures given at Harvard University. Second edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 1992. viii+168 pp.]


