= Solution
Use a Hermitian <gauge field> $A_\mu=A_\mu^AT^A$ and $\operatorname{tr}(T^AT^B)=\delta^{AB}/2$. With fundamental <covariant derivative> $\partial_\mu-igA_\mu$, the adjoint derivative is
$$
\boxed{D_\mu\Phi=\partial_\mu\Phi-ig[A_\mu,\Phi].}
$$
It transforms as $D_\mu\Phi\mapsto U(D_\mu\Phi)U^\dagger$, provided $A'_\mu=UA_\mu U^\dagger-(i/g)(\partial_\mu U)U^\dagger$.
The infinitesimal parameter in $U=1+i\epsilon+O(\epsilon^2)$ must be \b[Hermitian and traceless], from unitarity and unit determinant. It is not generally antisymmetric as the hint states. The valid generator $\epsilon=\operatorname{diag}(1,-1,0,0,0)$ is a concrete counterexample to that assertion.
Choose canonical adjoint kinetic normalization $\mathcal L_\Phi=\operatorname{tr}(D_\mu\Phi D^\mu\Phi)$, which gives $\tfrac12\partial\Phi^A\partial\Phi^A$. For the diagonal <vacuum expectation value> $D=v\operatorname{diag}(2,2,2,-3,-3)$, the matrix unit $E_{ij}$ obeys $[D,E_{ij}]=(d_i-d_j)E_{ij}$. Thus the <mass> of the <gauge boson> associated with either normalized Hermitian generator joining $i,j$ is $g|d_i-d_j|$.
Within the three-dimensional and two-dimensional blocks, or for diagonal generators, the commutator vanishes. There are therefore $8+3+1=12$ massless <gauge bosons>. The $3\times2$ interblock pairs supply two real generators each, hence 12 equally massive <gauge bosons>:
$$
\boxed{m_X=m_Y=5g|v|,\qquad
SU(5)\longrightarrow S(U(3)\times U(2))
\simeq\frac{SU(3)\times SU(2)\times U(1)}{\mathbb Z_6}.}
$$
The absolute mass formula assumes the stated kinetic normalization, which the question leaves implicit. Writing instead $\tfrac12\operatorname{tr}(D\Phi)^2$ without rescaling $\Phi$ would multiply these squared masses by $1/2$.
This is the gauge group of the <Standard Model>, up to its global quotient, arising from one adjoint <Higgs mechanism> in a simple unified group. The unbroken abelian generator can be chosen as $Y=\operatorname{diag}(-1/3,-1/3,-1/3,1/2,1/2)$, so $D=-6vY$. This is the characteristic first breaking step of an <SU(5) grand unified theory>, rather than electroweak breaking all the way to electromagnetism.
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