  Construction of superintegrable systems based on Lie algebras have been
introduced over the years. However, these approaches depend on explicit
realisations, for instance as a differential operators, of the underlying Lie
algebra. This is also the case for the construction of their related symmetry
algebra which take usually the form of a finitely generated quadratic algebra.
These algebras often display structure constants which depend on the central
elements and in particular on the Hamiltonian. In this paper, we develop a new
approach reexamining the case of the generic superintegrable systems on the
2-sphere for which a symmetry algebra is known to be the Racah algebra $R(3)$.
Such a model is related to the 59 $2D$ superintegrable systems on conformally
flat spaces and their 12 equivalence classes. We demonstrate that using further
polynomials of degree 2,3 and 4 in the enveloping algebra of $su(3)$ one can
generate an algebra based only on abstract commutation relations of $su(3)$ Lie
algebra without explicit constraints on the representations or realisations.
This construction relies on the maximal Abelian subalgebra, also called MASA,
which are the Cartan generators and their commutant. We obtain a new
6-dimensional cubic algebra where the structure constant are integer numbers
which reduce from a quartic algebra for which the structure constant depend on
the Cartan generator and the Casimir invariant. We also present other form of
the symmetry algebra using the quadratic and cubic Casimir invariants of
$su(3)$. It reduces as the known quadratic Racah algebra $R(3)$ only when using
an explicit realization. This algebraic structure describe the symmetry of the
generic superintegrable systems on the 2 sphere. We also present a contraction
to another 6-dimensional cubic algebra which would corresponding to the
symmetry algebra of a Smorodinsky-Winternitz model.
