Prove the \(S U(2)\) results in Eq. (5.3.28): \(\mathbf{2} \otimes \mathbf{2}=\mathbf{3} \oplus \mathbf{1} ; \mathbf{3} \otimes \mathbf{3}=\mathbf{5} \oplus \mathbf{3} \oplus \mathbf{1}\); and \(\mathbf{2 j}_{\mathbf{1}}+\mathbf{1} \otimes\) \(\mathbf{2} \mathbf{j}_{\mathbf{2}}+\mathbf{1}=\oplus_{J=\left|j_{1}-j_{2}\right|}^{j_{1}+j_{2}}(\mathbf{2 J}+\mathbf{1})\).