Polynomial Poisson Algebras and Superintegrable Systems from Cartan Centralisers of Types $$B_3$$, $$C_3$$ and $$D_3$$

Abstract In this work, we develop a root system construction for Cartan centralisers of classical complex semisimple Lie algebras of types B , C and D , and we analyse the associated polynomial Poisson algebras completely and explicitly in the rank-three cases $$B_3$$ B 3 , $$C_3$$ C 3 , and $$D_3$$ D 3 . This rank 3 case is already highly non-trivial. For $$B_n$$ B n , $$C_n$$ C n and $$D_n$$ D n , we use this new method to present the main division of cases and how the indecomposable polynomials organize into different types that can be understood via the root systems and their properties. An explicit description of all indecomposable polynomials of type $$B_n$$ B n is provided only for the rank 3. We provide insight how the type A construction changes once the root system contains signed long roots and, in types $$B_n$$ B n and $$C_n$$ C n , short or doubled roots. Thus, the directed-cycle construction for type $$A_{n-1}$$ A n - 1 becomes a sign-indexed factor construction with permutation-root, long-root, and short-root generators. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. The resulting finite rank-three sets of generators are used to describe the inclusion relations between the corresponding finitely generated Poisson $$\mathbb C[\mathfrak {h}^*]$$ C [ h ∗ ] -algebras and to construct algebraic superintegrable systems attached to the reduction chains $$\mathfrak {h} \subset \mathfrak {so}(6,\mathbb {C})$$ h ⊂ so ( 6 , C ) , $$\mathfrak {h} \subset \mathfrak {so}(7,\mathbb {C})$$ h ⊂ so ( 7 , C ) , and $$\mathfrak {h}