I’ve been going through Lee’s Introduction to Differential Geometry. It’s a great book but I feel it lacks examples and concrete applications of the ideas presented. I want to work out a list of problems involving quotients of Lie group actions, in particular quotients by normal subgroups.

I want to go beyond classical examples like: $\mathbb{R}P^n = \mathbb{R}^{n+1}/\mathbb{R}^\times$, $SO(\mathbb{R}, 3) = SU(\mathbb{C}, 2)/C_2$, $\mathbb{R}^\times = GL(\mathbb{R}^n)/SL(\mathbb{R}^n)$... These are all way too easy. Also, it would be nice to have examples where the manifolds aren't naturally embedded in $\mathbb{R}^n$ so that I'm forced to use the charts to run all the computations.

I’d be specially interested in objects that have some historical relevance or cases where the quotient is used as a tool to solve a concrete problem.