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Let $\beta$ be an involution on $\mathbb{C}[x,y]$, namely, $\beta$ is a $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ of order two. Denote the set of symmetric elements with respect to $\beta$ by $S_{\beta}$ and the set of skew-symmetric elements w.r.t. $\beta$ by $K_{\beta}$.

Let $s_1,s_2 \in S_{\beta}$ and $k_1,k_2 \in K_{\beta}$. Assume that every two of $\{s_1,s_2,k_1,k_2\}$ are algebraically independent over $\mathbb{C}$. Then $\mathbb{C}(s_1,s_2,k_1,k_2)$ has transcendence degree two over $\mathbb{C}$.

By a known result from algebraic geometry (which appears here), there exist $u,v \in \mathbb{C}(x,y)$ such that $\mathbb{C}(s_1,s_2,k_1,k_2)=\mathbb{C}(u,v)$.

Assume, for simplicity, that $u,v \in \mathbb{C}[x,y]$ (perhaps it is possible to find such $u,v$, see this question).

Call Property P the following property: There exist an automorphism $g$ of $\mathbb{C}[x,y]$ such that $g(u),g(v) \in S_{\beta} \cup K_{\beta}$.

Could one find a (counter)example in which property P is not satisfied? Or a proof that property P is satisfied?

This question seems slightly relevant (perhaps we should assume that some field extension here is Galois?).

Thank you very much!

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