Restrictions of representations of compact connected semisimple Lie groups to $SU(2)$ subgroups

Question

I would like to ask for some references for the following branching problem:

Let $G$ be a compact connected semisimple Lie group, and let $H$ be a closed Lie subgroup which is isomorphic to a copy of $SU(2)$. Given an irreducible representation space $V$ of $G$, describe the decomposition of $V$, as a representation space of $H$ by restriction, into irreducible representations of $H$.

I am more interested in the result itself, preferably using the language of weights and roots.

Edit: I have just learned about Kostant's branching theorem, which answers my question and more. One can find it for instance in Knapp's book "Lie Groups beyond an introduction", or of course, in Kostant's original work. It is sad that Kostant passed away recently, but he left behind many great results!

Answer 1

Kostant's branching theorem answers my question, and more.