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We reexamine different examples of reduction chains $\mathfrak{g} \supset\mathfrak{g}'$ of Lie algebras in order to show how the polynomials determiningthe commutant with respect to the subalgebra $\mathfrak{g}'$ leads topolynomial deformations of Lie algebras. These polynomial algebras have alreadybeen observed in various contexts, such as in the framework of superintegrablesystems. Two relevant chains extensively studied in Nuclear Physics, namely theElliott chain $\mathfrak{su}(3) \supset \mathfrak{so}(3)$ and the chain$\mathfrak{so}(5) \supset \mathfrak{su}(2) \times \mathfrak{u}(1)$ related tothe Seniority model, are analyzed in detail from this perspective. We show thatthese two chains both lead to three-generator cubic polynomial algebras, aresult that paves the way for a more systematic investigation of nuclear modelsin relation to polynomial structures arising from reduction chains. In order toshow that the procedure is not restricted to semisimple algebras, we also studythe chain $\hat{S}(3) \supset \mathfrak{sl}(2,\mathbb{R}) \times\mathfrak{so}(2)$ involving the centrally-extended Schr{\"o}dinger algebra in$(3+1)$-dimensional space-time. The approach chosen to construct thesepolynomial algebras is based on the use of the Lie-Poisson bracket, so that allthe results are presented in the Poisson (commutative) setting. The advantageof considering this approach {\it vs.} the enveloping algebras is emphasized,commenting on the main formal differences between the polynomial Poissonalgebras and their noncommutative analogue. As an illustrative example of thelatter, the three-generator cubic algebra associated to the Elliott chain isreformulated in the Lie algebraic (noncommutative) setting.

Polynomial algebras from Lie algebra reduction chains $\mathfrak{g} \supset \mathfrak{g}'$