Matrix Elements and the Wigner Eckhart Theorem Example

Let ${\vec {J}}_{1}$ and ${\vec {J}}_{2}$ be two angular momentum operators, ${\vec {J}}={\vec {J}}_{1}+{\vec {J}}_{2}$ is the sum of these two vectors, and $|JM\rangle$ denotes the eigen states of ${\vec {J}}^{2}$ and ${\mathbf {J}}_{z}$.