  We provide a classification and explicit formulas for the elements that span
the centralizer of Cartan subalgebras of complex semisimple Lie algebras of
non-exceptional type in their universal enveloping algebra, and show that these
generate polynomial (Poisson) algebras. The precise structure for the case of
rank three simple Lie algebras is provided, and the inclusion relations between
the corresponding polynomial algebras illustrated. We develop the ideas of
constructing algebraic superintegrable systems and their integrals from the
generators of the centralizer subalgebras and their symmetry algebra. In
particular, we present explicitly one example the superintegrable system
corresponding to the Cartan centralizer of $\mathfrak{so}(6,\mathbb{C})$.
