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

In this work, we construct explicit formulas for the generators of the Cartan centralisers of complex semisimple Lie algebras $B_n,C_n$ and $D_n$, the case $A_n$ being already known \cite{campoamor2023algebraic}. The precise structures for the cases of rank-three simple Lie algebras ($B_3,C_3$ and $