The Casimir elements of the Racah algebra

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\mathfrak{R}$ 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 $$[A,B]=[B,C]=[C,A]=2D$$ 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 $\mathfrak{R}$. Additionally the element $\delta=A+B+C$ is central in $\mathfrak{R}$. The algebra $\mathfrak{R}$ was introduced by Genest-Vinet-Zhedanov. We consider a mild change in their setting to call each element in \begin{equation*} D^2+A^2+B^2 +\frac{(\delta+2)\{A,B\}-\{A^2,B\}-\{A,B^2\}}{2} +A (\beta-\delta) +B (\delta-\alpha)+\mathfrak{C} \end{equation*} a Casimir element of $\mathfrak{R}$, where $\mathfrak{C}$ is the commutative subalgebra of $\mathfrak{R}$ generated by $\alpha$, $\beta$, $\gamma$, $\delta$. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of $\mathfrak{R}$: \begin{align*} \Omega_A = D^2 + \frac{B A C +C A B}{2} + A^2 +B \gamma -C \beta -A \delta, \Omega_B = D^2 + \frac{C B A +A B C}{2} + B^2 +C \alpha -A \gamma -B\delta, \Omega_C = D^2 + \frac{A C B +B C A}{2} + C^2 +A \beta -B\alpha -C\delta. \end{align*} The set $\{\Omega_A,\Omega_B,\Omega_C\}$ is invariant under a faithful $D_6$-action on $\mathfrak{R}$. Moreover we show that any Casimir element $\Omega$ is algebraically independent over $\mathfrak{C}$; if ${\rm char\,}\mathbb{F}=0$ then the center of $\mathfrak{R}$ is $\mathfrak{C}[\Omega]$.