In quantum mechanics, we construct all the representations of the $SU(2)$ algebra $$[J_j,J_k]=i\varepsilon_{jk\ell}J_\ell$$ in the $|j,m\rangle$ basis. The allowed values of $j$ turn out to be $$j=0,\frac{1}{2},1,\frac{3}{2},\ldots$$ which are all said to be irreducible.
Is there a theorem that we can invoke to conclude that these representations are all irreducible?