= Solution
The irreducible finite-dimensional <unitary representations> of $SU(2)$ are $E_n=\operatorname{Sym}^n(\mathbb C^2)$, one for each $n\geq0$, 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 $[J_x,J_y]=iJ_z$ cyclically. The actual <SU(2) Lie algebra> generators are $L_a=-iJ_a$, so $[L_x,L_y]=L_z$. On an orthonormal weight basis $v_m$, $m=-j,-j+1,\ldots,j$, the complexified action is
$$
J_zv_m=mv_m,\qquad
J_\pm v_m=\sqrt{(j\mp m)(j\pm m+1)}\,v_{m\pm1},\quad J_\pm=J_x\pm iJ_y.
$$
The <Casimir operator> in this normalization is $C=-\sum_aL_a^2=\sum_aJ_a^2$, acting by
$$
\boxed{C|_{E_n}=j(j+1)I=\frac{n(n+2)}4I.}
$$
Other Lie-algebra metric normalizations rescale this scalar. On the diagonal subgroup $\operatorname{diag}(e^{it},e^{-it})$, the weights of $E_n$ are $n,n-2,\ldots,-n$, all with multiplicity one.
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