A note on smooth $S{L}_2$-surfaces

Working over a field $k$ of characteristic zero, we study the ring $\mathfrak{R}={\mathfrak{D}}^{{\mathbb{Z}}_2}$ where $\mathfrak{D}=k[x_0,x_1,x_2]\mathrm{/}(2x_0{x}_2-x_1^2-1)$ and ${\mathbb{Z}}_2$ acts by $x_i\to -x_i$. $\mathfrak{D}$ admits an algebraic $S{L}_2(k)$-action which restricts to $\mathfrak{R}$. Our results include the following. (1) If $k$ is algebraically closed, the smooth $S{L}_2$-surface $X=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathfrak{R})$ admits an algebraic embedding in ${\mathbb{A}}_k^4$, and for any such embedding the $S{L}_2(k)$-action on $X$ does not extend to ${\mathbb{A}}_k^4$. In addition, there is no algebraic embedding of $X$ in ${\mathbb{A}}_k^3$. (2) The automorphism group ${\mathrm{A}\mathrm{u}\mathrm{t}}_k(\mathfrak{R})$ acts transitively on the set of irreducible locally nilpotent derivations of $\mathfrak{R}$. (3) Every automorphism of $\mathfrak{R}$ extends to $\mathfrak{D}$, and ${\mathrm{A}\mathrm{u}\mathrm{t}}_k(\mathfrak{R})=PS{L}_2(k){\ast }_{H}T$ where $T$ is its triangular subgroup. (4) $\mathfrak{R}$ is non-cancellative, i.e., there exists a ring $\mathfrak{S}$ such that ${\mathfrak{R}}^{[1]}{\cong }_k{\mathfrak{S}}^{[1]}$ but $\mathfrak{R}\mathrm{̸}{\cong }_k\mathfrak{S}$. In order to distinguish $\mathfrak{R}$ from $\mathfrak{S}$, we calculate the plinth invariant for $\mathfrak{R}$.