 Solution

For eigenvalues $(v,-v,0)$, all entries are distinct. The $12$ difference has magnitude $2|v|$, while both the $13$ and $23$ differences have magnitude $|v|$. The adjoint scalar gauge-boson mass matrix therefore givesOnly the two diagonal generators commute with the vacuum. Its unbroken group is the maximal torusThe two real fields belonging to each $ij$ pair combine into oppositely charged conjugate vector states with equal mass. The further equality between the $13$ and $23$ masses comes from the equally spaced vacuum eigenvalues; the surviving Abelian group alone does not force distinct root-charge sectors to have equal masses. Thus a fourfold massive degeneracy occurs in both examples, but its relation to the unbroken continuous symmetry differs. The scalar potential has not been specified, so no additional symmetry of that potential or physical scalar mass spectrum can be inferred.