Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising (semisimple or non-semisimple) Lie $\mathfrak{g}$. In this approach, we deal with explicit polynomials in the enveloping algebra of $\mathfrak{g} \oplus \mathfrak{g} \mathfrak{g}$. present examples show how these can display different behaviours and lead Abelian algebras, quadratic more complex structures involving higher order nested commutators. Within framework, also demonstrate virtual copies Levi factor decomposable be used as tool "copies" algebras. Different schemes obtain associated algebraic Hamiltonians have been proposed literature, among them use commutants various type. The approach is relies on construction $\mathcal{U}(\mathfrak{g} \mathfrak{g})$.