Reference request: Conjugacy classes of real semi-simple Lie group $G$ and their decomposition into $K$-orbits.

Question

Let $G$ be a real semi-simple Lie group and $K$ its maximal compact subgroup.

1. Is there a description of the orbit space of the conjugation action on $G$ by itself?

2. Is there a decription of how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$?

I would appreciate it if anyone could point out any references.