The notes for my class on Kaluza-Klein reduction are a bit all over the place and at times it's difficult to follow what's going on. (I plan on asking a specific question about an example later).
For now, are there any resources that introduce Kaluza-Klein reduction with some worked examples, apart from the simple Minkowski space with a circle attached. I'd like to see some explicit examples where we have a higher dimensional torus for example.
A treatment including supergravity would be a plus.