Centralizer of the standard SU2 subgroup in SU3 ($C_{SU(3)}(SU(2))=\{\mathrm{diag}(z,z,z^{-2}):|z|=1\}$)

Embed SU(2) into SU(3) by $U\mapsto \mathrm{diag}(U,1)$. Its defining representation decomposes as an irreducible doublet and a trivial representation. The Schur lemma forces every commuting matrix to be $\mathrm{diag}(z{I}_2,w)$. Unitarity and unit determinant imply $|z|=1$ and $w=z^{-2}$, proving the formula. With Hermitian generator $X=\mathrm{diag}(1,1,-2)$, the defining representation restricts as $2_1\oplus 1_{-2}$ and the Adjoint representation as $3_0\oplus 2_3\oplus 2_{-3}\oplus 1_0$. The U(1) charges in the Adjoint representation are differences of defining charges. The generated connected subgroup is $(SU(2)\times U(1))/{\mathbb{Z}}_2$, since the two factors meet in $\mathrm{diag}(-I_2,1)$.