I am trying to understand the proof of theorem 1 from this paper V. Kac and B. Weisfeiler.

Here $Z$ is the center of universal enveloping algebra $U(G)$ (here $G = \operatorname{Lie} \mathscr{G}$ ). $T$ is a Lie algebra of a maximal torus. $Z^{ \mathscr{G} }$ denote invariants (with respect to Ad-action of group $\mathscr{G}$) in $Z$.

Proof is not very short. But the last two lines are these

Here $\bar{A}$ means field of fraction of ring $A$. Suppose I believe that these fields are isomorphic and these rings are integrally closed.

Question How does it imply isomorphism of initial rings?

For example $ \mathbb{k} [x, xy] \subset \mathbb{k} [x, y]$

I am trying to understand the proof of Theorem 1 from this paper V. Kac and B. Weisfeiler (Indag. Math. 1976, DOI link).

Theorem 1. Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(Z^\mathscr{G})=U(T)^W$ and $\gamma:Z^\mathscr{G}\to U(T)^W$ is a ring isomorphism.

Here $Z$ is the center of universal enveloping algebra $U(G)$ (here $G = \operatorname{Lie} \mathscr{G}$ ). $T$ is a Lie algebra of a maximal torus. $Z^{ \mathscr{G} }$ denote invariants (with respect to Ad-action of group $\mathscr{G}$) in $Z$.

Proof is not very short. But the last two lines are these

From 4.4, 4.6 and 5.3 we get $\overline{U(T)^W}=\overline{\gamma(Z^\mathscr{G})}$. Since both $U(T)^W$ and $\gamma(Z^\mathscr{G})$ are integrally closed we get $U(T)^W=\gamma(Z^\mathscr{G})$ as desired.

Here $\bar{A}$ means field of fraction of ring $A$. Suppose I believe that these fields are isomorphic and these rings are integrally closed.

Question How does it imply isomorphism of initial rings?

For example $ \mathbb{k} [x, xy] \subset \mathbb{k} [x, y]$