We continue the study in Ben-Shimol [1],[2] and consider a Borel subalgebra $\mathfrak{ b}$ and its nil radical $\mathfrak{ n}$ of the simple Lie algebras of types $G_2$, $F_4$, $C_n$ over arbitrary field. Let $\mathcal{ L}\in\{ \mathfrak{ n}, \mathfrak{ b}\}$. We establish here explicit realizations of the center $Z(\mathcal{ L})$ and semi-center $Sz(\mathcal{ L})$ of the enveloping algebra, the Poisson center $S(\mathcal{ L})^{ \mathcal{ L}}$ and Poisson semi-center $S(\mathcal{ L})^{ \mathcal{ L}}_{ \operatorname{ si}}$ of the symmetric algebra. We describe their structure as commutative rings and establish isomorphisms $Z(\mathcal{ L})\cong~S(\mathcal{ L})^{ \mathcal{ L}}$, $Sz(\mathcal{ L})\cong S(\mathcal{ L})^{ \mathcal{ L}}_{ \operatorname{ si}}$.
Categories: math.RT math.RA