 Centralizer of a Hermitian matrix in SU(n)

Let a Hermitian matrix have distinct real eigenvalues with multiplicities $n_1,\dots ,n_k$. A unitary matrix commuting with it must preserve each eigenspace: if $Av=\lambda v$, then $A(Uv)=UAv=\lambda Uv$. Conversely, any unitary transformation acting independently on these mutually orthogonal eigenspaces commutes with $A$. The centralizer in $U(n)$ is therefore ${\prod }_{j}U(n_j)$. Imposing total determinant one gives the displayed subgroup inside the special unitary group $SU(n)$.

For multiplicities $(2,1)$ in SU(3), it is $S(U(2)\times U(1))\cong U(2)$, embedded by $U\mapsto \mathrm{diag}(U,(\det U{)}^{-1})$. Three distinct eigenvalues leave the diagonal maximal torus $U(1{)}^2$. This determines the unbroken group for a diagonal adjoint scalar field vacuum, including its global determinant constraint.