  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 $\mathfrak{C}[\Omega]$.
