  We discuss a procedure to determine finite sets $\mathcal{M}$ within the
commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie
algebra $\mathfrak{g}$ such that their generators define a quadratic algebra.
Although independent from any realization of Lie algebras by differential
operators, the method is partially based on an analytical approach, and uses
the coadjoint representation of the Lie algebra $\mathfrak{g}$. The procedure,
valid for non-semisimple algebras, is tested for the centrally extended
Schr\"odinger algebras $\widehat{S}(n)$ for various different choices of
algebraic Hamiltonian. For the so-called extended Cartan solvable case, it is
shown how the existence of minimal quadratic algebras can be inferred without
explicitly manipulating the enveloping algebra.
