 Solution

The irreducible finite-dimensional unitary representations of $SU(2)$ are $E_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$, one for each $n\ge 0$, with dimension $n+1$. Equivalently they are spin-$j$ representations with $j=n/2$. The defining representation has $n=1$, and $n=0$ is trivial. Every finite-dimensional unitary representation is an orthogonal direct sum of these.

Choose self-adjoint angular-momentum generators $J_x,J_y,J_z$ with $[J_x,J_y]=i{J}_z$ cyclically. The actual SU(2) Lie algebra generators are $L_a=-i{J}_a$, so $[L_x,L_y]=L_z$. On an orthonormal weight basis $v_m$, $m=-j,-j+1,\dots ,j$, the complexified action isThe Casimir operator in this normalization is $C=-{\sum }_a{L}_a^2={\sum }_a{J}_a^2$, acting byOther Lie-algebra metric normalizations rescale this scalar. On the diagonal subgroup $\mathrm{diag}(e^{it},e^{-it})$, the weights of $E_n$ are $n,n-2,\dots ,-n$, all with multiplicity one.