Let $f,g \in \mathbb{C}[x,y]$. There is a well-known result, that can be found for example here, pages 19-20, that says the following:

$f,g$ are algebraically dependent over $\mathbb{C}$ if and only if their Jacobian $Jac(f,g):=f_xg_y-f_yg_x$ is zero.

Actually, this result is valid for $f_1,\ldots,f_n \in \mathbb{C}[x_1,\ldots,x_n]$, any $n \in \mathbb{N}$.

Now, let $f,g \in A_1(\mathbb{C})$, where $A_1(\mathbb{C})$ is the first Weyl algebra over $\mathbb{C}$, namely, the $\mathbb{C}$-algebra generated by $x,y$ such that $[y,x]=yx-xy=1$.

I wonder if there exists an analog result to the above in $A_1(\mathbb{C})$, namely:

$f,g \in A_1(\mathbb{C})$ are 'algebraically dependent' over $\mathbb{C}$ if and only if $[f,g]=0$.

One has to be careful because:

(1) One has to define algebraic dependence over $\mathbb{C}$ of two non-commuting elements $f$ and $g$. Should it be $\sum \lambda_{ij}f^ig^j=0$, with $\lambda_{ij} \in \mathbb{C}$ not all zero, or $\sum \lambda_{ij}f^ig^j + \sum \mu_{ij}g^if^j=0$, with $\lambda_{ij}, \mu_{ij} \in \mathbb{C}$ not all zero?. (Perhaps the first definition should be called 'one-sided algebraic dependence', while the second definition should be called 'two-sided algebraic dependence').

(2) Perhaps this question is relevant. The example there (of Dixmier) is of $U,V \in A_1$, $[U,V]=0$, and $U^3-V^2+1=0$, so it does not contradict my plausible analog result, since those $U$ and $V$ are algebraically dependent.

((3) I am not sure if this is relevant, but in the above mentioned reference, on page 11, the Gelfand-Kirillov dimension is mentioned with connection to transcendence degree; for $\mathbb{C}[x,y]$ those notions coincide. Is the fact that the Gelfand-Kirillov dimension of $A_1(\mathbb{C})$ is two relevant to my question?).

Thank you very much for any help!