A principal $G$-bundle has a cross section iff it is trivial (e.g. Husemoller's Fibre Bundles, 3rd ed., 8.3 in chapter 4).
A principal $G$-bundle is in particular a fiber bundle with fiber $G$.
My question: does there exist a group $G$ and a non-trivial principal $G$-bundle $p:E\rightarrow B$ that does have a cross section when considered as a mere fiber-bundle?
If so, I would be glad to see a simple example. Thanks!