Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$. Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$. Similarly, define $t_y(E)$ to be the maximum among $0$ and the $y$-degree of $E(0,y)$.

The following nice result appears in several places, for example: Proposition 2.1 Lemma 1.14 Proposition 10.2.6.

Nice result: Let $A,B \in k[x,y]$ satisfy $\operatorname{Jac}(A,B) \in k-\{0\}$ (such $A,B$ is called a Jacobian pair). Assume that the $(1,1)$-degree of $A$, $\deg(A)$, is $>1$ and the $(1,1)$-degree of $B$, $\deg(B)$, is $>1$. Then the numbers $t_x(A),t_y(A),t_x(B),t_y(B)$ are all positive.

My question: Is the same result holds in the first Weyl algebra over $k$, $A_1(k)$? where instead of the Jacobian we take the commutator.

Of course, we must first define $t_x(A),t_y(A),t_x(B),t_y(B)$ in $A_1(k)$; it seems to me that the same definition holds for $A_1(k)$, or am I missing something? Perhaps it is not possible to consider $E(x,0)$, where $E \in A_1(k)$?

If I am not wrong, the proof of Proposition 2.1 can be adjusted to the non-commutative case:

(i) It is easy to see that Lemma 1.3 has a non-commutative analog.

(ii) Replacing the Jacobian by the commutator yields a similar result (use $[ab,c]=a[b,c]+[a,c]b$, and then the same conclusion.

I have asked this question here.

Thank you very much!