From Lemma 1.1.8, we obtain the following: Assume that $R \subseteq S$ are commutative rings and $u: R[x,y] \to R[x,y]$ is an $R$-algebra endomorphism that has an invertible Jacobian, namely, $Jac(u(x),u(y)) \in R^*$. Denote $A:=u(x) \in R[x,y]$, $B:=u(y) \in R[x,y]$ and let $v: S[x,y] \to S[x,y]$ be the $S$-algebra endomorphism defined as follows: $v: (x,y) \mapsto (A,B)$ (and extended to $S[x,y]$ in the obvious way). If $v$ is an automorphism of $S[x,y]$, then $u$ is an automorphism of $R[x,y]$.
Denote the first Weyl algebra over $R$ by $A_1(R)$; by definition it is the $R$-algebra generated by $x$ and $y$ such that $yx-xy=1$.
Is it possible to replace $R[x,y]$ and $S[x,y]$ by $A_1(R)$ and $A_1(S)$?
Namely, assume that $u: A_1(R) \to A_1(R)$, $u: (x,y) \mapsto (A,B)$, $[B,A]=1$, and let $v: A_1(S) \to A_1(S)$, $v: (x,y) \mapsto (A,B)$. If $v$ is an automorphism of $A_1(S)$, then $u$ is an automorphism of $A_1(R)$.
Remarks: (1) One has to be careful, since perhaps $A_1(R)$ or $A_1(S)$ are not simple rings (I do not know for which commutative rings $T$, $A_1(T)$ are simple; see this question). (2) Further assume that $R$ and $S$ are integral domains of characteristic zero. Probably a counterexample to my question is a counterexample to the Dixmier Conjecture, so (since I 'believe' in the Dixmier Conjecture) I guess there is no counterexample. But how to prove the claim?
