I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups.
When a Lie group has a set of irreducible representations, I'd like to know how these representations decompose into irreducible representations of a subgroup.
I heard of "Symmetry, representations, and invariants" by Goodman and Wallach and "Representation Theory" by Fulton and Harris, but I couldn't get an account on the special cases I'm interested in, which are $U(1) \to SU(2)$ and $SO(4) \to SO(5)$. I know that $U(1) \cong \operatorname{Spin}(2, \mathbb{R})$ and $SU(2) \cong \operatorname{Spin}(3, \mathbb{R})$, but Goodman/Wallach and Fulton/Harris only seem to treat $\operatorname{Spin}(n, \mathbb{C})$ and $SO(n, \mathbb{C})$.



What is a good introduction to branching rules in representation theory?
Zhelobenko has books on the subject from 1970, 1983, 1994, 2004. I'm pretty sure it's the 1970 that I saw the most concrete branching laws in.


The special cases you're interested in are *really* easy, by the way, in that the branching laws are "multiplicity-free". For example, the $SU(2)$ irrep of dimension $n+1$ breaks into the $U(1)$ irreps with weights $n,n-2,n-4,\ldots,-n$, each only once.


