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For $n\ge 2$, contraction with the invariant symmetric bilinear form splits off the trivial line in the symmetric square of the defining special orthogonal group representation. Its null space is the traceless symmetric square. It contains a highest-weight vector of weight $2e_1$, and the Weyl dimension formula gives $(n+1)(2n-1)$, exactly its dimension. Complete reducibility of compact-group representations therefore makes that null space irreducible. The entire symmetric square is reducible because of the invariant line.

For active rotations with $U=e^{-i\varphi J_3}{e}^{-i\theta J_2}{e}^{-i\psi J_3}$, the irreducible spin representation matrix is $D_{m{m}^{\prime }}^{(j)}=e^{-im\varphi }{d}_{m{m}^{\prime }}^{(j)}(\theta )e^{-i{m}^{\prime }\psi }$, where $d_{m{m}^{\prime }}^{(j)}=⟨j,m|e^{-i\theta J_2}|j,m^{\prime }⟩$. Its spin-one-half middle matrix is $\left(\begin{array}{cc}\cos (\theta /2)& -\sin (\theta /2)\\ \sin (\theta /2)& \cos (\theta /2)\end{array}\right)$. A $2\pi$ rotation acts by $(-1{)}^{2j}$; only integer-spin representations descend to the special orthogonal group in three dimensions.