This question is about an example of interpreting a field in an affine group, from Section 1.3 of Marker's Model theory: An introduction.
Let $F$ be an infinite field and $G$ be the group of transformation of the form $x \mapsto ax + b$, where $a \in F^\times$ and $b \in F$ (write $T(a, b)$ for such a transformation). The aim is to show that $F$, as a structure in the language of rings, is definably interpretable in $G$, as a structure in the language of groups.
It is easy to identify the subgroups of $G$ that are suitable for interpreting multiplication and addition, resp., namely: $$ B := \{T(x, 0)\}\ \text{and}\ A := \{T(1, x)\} $$
Then Marker makes two crucial points for the definability, by using group-theoretic devices. The first is that the map $i:A \setminus \{1\}\rightarrow B$; $T(1, x) \rightarrow T(0, x)$ is definable by using the action of $B$ on $A$ by conjugation: $i(a) = b$ iff $bab^{-1} = \alpha$, where $\alpha = T(1, 1)$. The second is that the subgroups $A$ and $B$ are also definable: they are the groups of elements that commutes with $\alpha$ and $\beta$, resp., where $\beta = T(\tau, 0)$ and $\tau \neq 0, 1$.
My question is how to come up with these facts. I'm surprised by them because what I naturally want to define turn out to be just centralizers or a map obtained from conjugation. Are these just ad-hoc arguments, or are there a general (group-theoretic) principles that make $i, A$ and $B$ definable?