How do I take a tensor products of $SO(N)$ irreps and decompose it in terms of irreps for $N>3$? (I understand the special case of $SO(3)$ we can use the nice $SU(N)$ technology of Young Tableauxs but I am not interested in that case).
Are there general procedures? If not are there procedures that work for specific representations (fundamental, adjoint ,something else...)?
I already found this this very useful result in another answer (last part). This says that the fundamental representation always can be decomposed in terms of traceless, symmetric traceless and antisymmetric representations for all $SO(N)$. If we stick with just fundamental for a second what would happen when we take the tensor product with a third (and forth etc...) fundamental representation?
I am not that interested in $SO(6)$ results as it is basically similar to $SO(3)$ in that you can just use $SU(N)$ techniques. I am more interested in what you can do when there is no isomorphism to $SU(N)$.