  The notion of hidden symmetry algebra used in the context of exactly solvable
systems is re-examined from the purely algebraic way, analyzing subspaces of
commuting polynomials that generate finite-dimensional quadratic algebras. By
construction, these algebras do not depend on the choice of realizations by
vector fields of the underlying Lie algebra, allowing to propose a procedure to
analyze polynomial algebras as those subspaces in an enveloping algebra that
commute with a given algebraic Hamiltonian.
