= Adjoint scalar gauge-boson mass matrix
{title2=$(m^2)^{ab}=-2g^2\operatorname{tr}([t^a,\Phi_0][t^b,\Phi_0])$}
For a Hermitian <adjoint scalar field> with kinetic term $\operatorname{tr}(D_\mu\Phi(D^\mu\Phi)^\dagger)$, use $D_\mu\Phi=\partial_\mu\Phi+ig[A_\mu,\Phi]$ and generators normalized by $\operatorname{tr}(t^at^b)=\delta^{ab}/2$. Expanding at a constant <vacuum expectation value> $\Phi_0$ and matching $\mathcal L_{\rm mass}=A_\mu^a(m^2)^{ab}A^{b\mu}/2$ gives the displayed expression. A real combination of generators $X=x_at^a$ has quadratic form $2g^2\operatorname{tr}([X,\Phi_0]^\dagger[X,\Phi_0])\ge0$. Thus the massless directions are exactly the generators commuting with the vacuum.
For $\Phi_0=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$, the two Hermitian generators associated with each <matrix unit> pair $E_{ij},E_{ji}$ have squared <mass> $g^2(\lambda_i-\lambda_j)^2$. The surviving group is the <centralizer of a Hermitian matrix in SU(n)>.
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