Use \(\mathrm{SO}(3)\) group characters to show that in Eq. (6.44) the coefficients \(c_{J}\) are all zero or one [so \(\mathrm{SO}(3)\) is simply reducible], which leads to the \(\mathrm{SO}(3)\) Clebsch-Gordan series (6.45). Use the results of Section 6.3 .4, and that the \(\mathrm{SO}(3)\) characters \(\chi_{\alpha}^{(j)}\) obey a relation \(\chi_{\alpha}^{\left(j_{1}\right)} \chi_{\alpha}^{\left(j_{2}\right)}=\sum_{J} c_{J} \chi_{\alpha}^{(J)}\) that is analogous to Eq. (6.44).