How to prove that the massive gauge fields transform in the fundamental representation of the gauge group of unbroken generators?

In Chapter 84 (page 528) of Srednicki’s Quantum Field Theory, he wrote

Recall from section $32$ that a generator $T^a$ is spontaneously broken if $(T^a_{scriptscriptstyle{text{R}}})_{ij} v_j ne 0$. From eq.$(84.11)$, we see that gauge fields correponding to broken generators get a mass, while those corresponding to unbroken generators do not. The unbroken generators (if any) form a gauge group with massless gauge fields. The massive gauge fields (and all other fields) form representations of this unbroken group.

Let us work out some simple examples.

Consider the gauge group $SU(N)$, with a complex scalar field $varphi$ in the fundamental representation. We can make a global $SU(N)$ transformation to bring the VEV entirely into the last component, and furthermore make it real. Any generator ${(T^a)_i}^j$ that does not have a nonzero entry in the last column will remain unbroken. These generators form an unbroken $SU(N-1)$ gauge group. There are three classes of broken generators: those with ${(T^a)_i}^N =frac{1}{2}$ for $i ne N$ (there are $N-1$ of these); those with ${(T^a)_i}^N = -frac{1}{2}i$ for $ine N$ (there are also $N-1$ of these), and finally the single generator $T^{N^2-1}=[2N(N-1)]^{-1/2} text{diag}(1,cdots,1,-(N-1))$. The gauge fields corresponding to the generators in the first two classes get a mass $M=frac{1}{2}gv$; we can group them into a complex vector field that transforms in the fundamental representation of the unbroken $SU(N-1)$ subgroup. The gauge field corresponding to $T^{N^2-1}$ gets a mass $M=[(N-1)/2N]^{1/2}gv$; it is a singlet of $SU(N-1)$.

where I’ve marked as bold the sentences I can’t understand.

How to prove that the massive gauge fields transform in the fundamental representation of the gauge group of unbroken generators?

Edit:

In order to avoid ambiguity, I’ve modified the question from ‘how to understand’ to ‘how to prove’.