The universal enveloping algebra of $\mathfrak{sl}_2$ and the Racah algebra

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\Re$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$. The relations assert that \begin{equation*} [A,B]=[B,C]=[C,A]=2D \end{equation*} and each of the elements \begin{gather*} \alpha=[A,D]+AC-BA, \qquad \beta=[B,D]+BA-CB, \qquad \gamma=[C,D]+CB-AC \end{gather*} is central in $\Re$. Additionally the element $\delta=A+B+C$ is central in $\Re$. In this paper we explore the relationship between the Racah algebra $\Re$ and the universal enveloping algebra $U(\mathfrak{sl}_2)$. Let $a,b,c$ denote mutually commuting indeterminates. We show that there exists a unique $\mathbb{F}$-algebra homomorphism $\natural:\Re\to\mathbb{F}[a,b,c]\otimes_\mathbb{F} U(\mathfrak{sl}_2)$ that sends \begin{eqnarray*} A &\mapsto& a(a+1)\otimes 1+(b-c-a)\otimes x+(a+b-c+1)\otimes y-1\otimes xy, \\ B &\mapsto& b(b+1)\otimes 1+(c-a-b)\otimes y+(b+c-a+1)\otimes z-1\otimes yz, \\ C &\mapsto& c(c+1)\otimes 1+(a-b-c)\otimes z+(c+a-b+1)\otimes x-1\otimes zx, \\ D &\mapsto& 1\otimes (zyx+zx)+ (c+b(c+a-b))\otimes x +(a+c(a+b-c))\otimes y \\ && \qquad+(b+a(b+c-a))\otimes z +\,(b-c)\otimes xy+(c-a)\otimes yz+(a-b)\otimes zx, \end{eqnarray*} where $x,y,z$ are the equitable generators for $U(\mathfrak{sl}_2)$. We additionally give the images of $\alpha,\beta,\gamma,\delta,$ and certain Casimir elements of $\Re$ under $\natural$. We also show that the map $\natural$ is an injection and thus provides an embedding of $\Re$ into $\mathbb{F}[a,b,c]\otimes U(\mathfrak{sl}_2)$. We use the injection to show that $\Re$ contains no zero divisors.