Angular momentum algebra, Wigner-Eckart theorem

We define an irreducible spherical tensor \( T^{\lambda}_{\mu} \) of rank \( \lambda \) as an operator with \( 2\lambda+1 \) components \( \mu \) that satisfies the commutation relations (\( \hbar=1 \)) $$ [J_{\pm}, T^{\lambda}_{\mu}]= \sqrt{(\lambda\mp \mu)(\lambda\pm \mu+1)}T^{\lambda}_{\mu\pm 1}, $$ and $$ [J_{z}, T^{\lambda}_{\mu}]=\mu T^{\lambda}_{\mu}. $$