Proposition 1

Let $K$ be a field. For every subset $X$ of $K$, the centralizer (§ 8, no. 5, Example 3) $X'$ of $X$ is a subfield of $K$.

We know (loc. cit.) that $X'$ is a subring of $K$. On the other hand, if $x \neq 0$ is permutable with $z \in X$, so is $x^{-1}$ (I, § 2, no. 3, Proposition 5) and hence $X'$ contains the inverse of every non-zero element of $X'$.