Let $G$ be a Lie group (possibly disconnected). Consider the natural short-exact sequence $$1\rightarrow G\_0\rightarrow G\rightarrow\pi\_0(G)\rightarrow 1,$$ where $G\_0$ is the identity component of $G$, and $\pi\_0(G)\cong G/G\_0$ is the component group. Under what conditions does this sequence admit a splitting, so that $G$ is a semidirect product of $G\_0$ and $\pi\_0(G)$?. Useful answers might include (but need not be limited to) some types of Lie group $G$ for which a splitting exists. I would appreciate any and all references and suggestions.


Thanks!



Splitting of a Short-Exact Sequence of Lie Groups
See pages 177 to 190 of:


Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. 
[(pdf)](http://www.mat.univie.ac.at/~michor/dgbook.pdf)


