Automorphisms of simple quotients of the Poisson and universal enveloping algebras of ${\mathrm{s}\mathrm{l}}_2$

Let $P({\mathrm{s}\mathrm{l}}_2(K))$ be the Poisson enveloping algebra of the Lie algebra ${\mathrm{s}\mathrm{l}}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $ $ $P({\mathrm{s}\mathrm{l}}_2(K))\mathrm{/}(C_P-\lambda )$, where $C_P$ is the standard Casimir element of ${\mathrm{s}\mathrm{l}}_2(K)$ in $P({\mathrm{s}\mathrm{l}}_2(K))$ and $0\neq \lambda\in K$, are proven to be simple in \cite{UZh}. Using a result by L. Makar-Limanov \cite{ML90}, we describe generators of the automorphism group of $P({\mathrm{s}\mathrm{l}}_2(K))\mathrm{/}(C_P-\lambda )$ and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by J. Dixmier \cite{Dixmier73} and O. Fleury \cite{Fleury} for the quotient algebras $U({\mathrm{s}\mathrm{l}}_2(K))\mathrm{/}(C_U-\lambda )$, where $C_U$ is the standard Casimir element of ${\mathrm{s}\mathrm{l}}_2(K)$ in the universal enveloping algebra $U({\mathrm{s}\mathrm{l}}_2(K))$, we prove that the automorphism groups of $P({\mathrm{s}\mathrm{l}}_2(K))\mathrm{/}(C_P-\lambda )$ and $U({\mathrm{s}\mathrm{l}}_2(K))\mathrm{/}(C_U-\lambda )$ are isomorphic.