  We review some aspects of the Racah algebra $R(n)$, including the closure
relations, pointing out their role in superintegrability, as well as in the
description of the symmetry algebra for several models with coalgebra symmetry.
The connection includes the generic model on the $(n-1)$ sphere. We discuss an
algebraic scheme of constructing Hamiltonians, integrals of the motion and
symmetry algebras. This scheme reduces to the Racah algebra $R(n)$ and the
model on the $(n-1)$ sphere only for the case of specific differential operator
realizations. We review the method, which allows us to obtain the commutant
defined in the enveloping algebra of $\mathfrak{sl}(n)$ in the classical
setting. The related $\mathcal{A}_3$ polynomial algebra is presented for the
case $\mathfrak{sl}(3)$. An explicit construction of the quantization of the
scheme for $\mathcal{A}_3$ by symmetrization of the polynomial and the
replacement of the Berezin bracket by commutator and symmetrization of the
polynomial relations is presented. We obtain the additional quantum terms.
These explicit relations are of interest not only for superintegrability, but
also for other applications in mathematical physics.
