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 ${\mathbf {J}}_{z}$.

${\mathbf {(}}a)$ Show that the matrix elements of $J_{1}^{-},\langle JM|J_{1}^{-}|J'(M+1)\rangle$, vanish, unless ${\mathbf {J}}'={\mathbf {J}}$ or ${\mathbf {J}}'={\mathbf {J}}\pm 1$.

${\mathbf {(}}b)$ Show also that the following expressions are independent of ${\mathbf {M}}$: