I have a reducible SU(2) representation $D^{(\alpha)}$, which can be decomposed as, say $D^{(\alpha)} = D^{(m)}\oplus D^{(n)}$ where the representations on the right-hand side are irreducible. I know that there is a general formula for the decomposition of the tensor product of irreducible representations (see Wikipedia).
Now, if I want to find the decomposition of the tensor product of the reducible representation, i.e., $D^{(\alpha)} \otimes D^{(\alpha)}$, is it possible to write a general formula for the decomposition using the results of the Wikipedia entry and the decomposition of $D^{(\alpha)}$?