Representations of the rotation group

Question

(I have already done a similar question, but I did not express myself very well and the question was a bit confusing, so let me try again. If you find the question repetitive, I apologize and you can remove it). Consider a Lie group with 3 generators $J_{1}, J_{2}$ and $J_{3}$ that satisfy $$[J_{a},J_{b}] = i\epsilon_{abc} J_{c}.$$ We can show that $J^{2}$ commutes with all these generators, so we can diagonalize $J^{2}$ and one of them togheter (we choose $J_{3}$. If we call $|j,m \rangle$ a eigenvector of $J_{3}$ and $J^{2}$, we find that $$J^{2}|j,m \rangle = j(j+1)|j,m \rangle,$$ $$J_{3}|j,m \rangle = m |j,m \rangle,$$ where $j = 0, 1/2, 1, 3/2, ...$ and $m = -j, -j+1, ..., j$. I have read that for each value of $j$ is associated a representtion of this group, given by the matrices with elements $\langle j, m | J_{a} |j,m \rangle$. In fact, for $j = 1/2$ we obtein the Pauli matrices, which are related to the generators of $SU(2)$. But how can I show that this matrices form a representation? It's not obvious to me.

Answer 1

Simply show that the matrices satisfy the commutation relations that define the algebra. The commutator is defined in terms of matrix multiplication. For matrices $X,Y$, we define $[X,Y]=XY-YX$.