In Sakurai's book it's written that the operator $D_{m',m}^{(j)}=\left\langle{j,m'}\Big|\exp{\frac{-i \mathbf{ J\cdot \hat{n} } \phi}{\hbar}}\Big|{j,m}\right\rangle$ is the "$2j+1$-dimensional irreducible representation of the rotation operator". I was wondering whether there is a simple way to prove that it is irreducible (if it's not obvious for some reasons I'm missing).
For example, considering $j=1$ and the vector space generated by the eigenkets of $J^2$ and $J_z$, how can I show that the components of the angular momentum generate an irreducible representation?
My idea would be to use the fact that any other operator that commutes with $J_x$, $J_y$ and $J_z$ is a multiple of the identity (which is proved easily) and then use some sort of converse of Schur's lemma, if it even exists.
I'm not even sure this is the best way to tackle the problem, as I only know the most basic things about group theory.