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For eigenvalues $(v,v,-2v)$, the first two entries coincide. Consequently the $12$ generators $t^1,t^2$, as well as $t^3,t^8$, commute with the vacuum. The $13$ and $23$ eigenvalue differences both have magnitude $3|v|$, givingThere are four massless and four massive gauge bosons. The centralizer of a Hermitian matrix in SU(n) preserves the degenerate two-dimensional eigenspace, so the unbroken group isAt the Lie algebra level this is $\mathfrak{su}(2)\oplus \mathfrak{u}(1)$, generated by $t^{1,2,3}$ and $t^8$. The finite quotient specifies the actual subgroup inside SU(3); simply writing $SU(2)\times U(1)$ suppresses that global identification. The four real massive fields combine into a complex SU(2) doublet and its conjugate. Unbroken SU(2) rotations relate its two components, explaining the equal masses of the $13$ and $23$ sectors in this SU(3) gauge theory with an adjoint Higgs field.