Since the CG coefficient is already normalized.Thus

\displaystyle \sum_{m_1, m_2} \left(C^{j_1 m_1 j_2 m_2}_{JM}\right)^2 = 1

Since the number of M is 2J+1, as M = -J, -J+1, ... J . Thus,

\displaystyle \sum_M \sum_{m_1, m_2} \left(C^{j_1 m_1 j_2 m_2}_{JM}\right)^2 = 2J+1

At last, the number of dimension of the coupled space or (tensor product space) is equation to (2j_1 +1) (2j_2+1) , i.e.

\displaystyle \sum_J (2J+1) = (2j_1+1)(2j_2+1)

Thus,

\displaystyle \sum_{JM, m_1 m_2} \left(C^{j_1 m_1 j_2 m_2}_{JM}\right)^2 = (2j_1+1)(2j_2+1)

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