The reduced CG coefficients for $SO(d):SO(d-1)$ are in principle known in full generality for $d\leq 4$: they are trivial for $d=2$, equivalent to $3j$ symbols of $SU(2)$ for $d=3$, and to $9j$ symbols of $SU(2)$ for $d=4$. For some classes of tensor products (e.g multiplicity-free tensor products) the reduced CG coefficients seem to be more or less known for general $d$.
Are the general reduced CG coefficients (or the full CG coefficients) known for $SO(5):SO(4)$? This definitely does not fall into the multiplicity-free category since already the adjoint representation has weight multiplicities.
My motivation for this question comes from the fact that $SO(d)$ CG coefficients are useful for describing three-point correlation functions in $(d+1)$-dimensional quantum conformal field theories and 6d CFTs are a topic of active research.
