Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those functions $f:M_{2n}(\mathbb{R})\to \mathbb{R}$ with $f(AB)=f(BA)$, would be closed under the Poisson bracket ?

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb{R})\to \mathbb{R}$ with $f(AB)=f(BA)\;\;\;\; \forall A,B \in M_{2n}(\mathbb{R})$, would be closed under the Poisson bracket ?

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficient, such that the space of smooth invariant functions, those functions $f:M_{2n}(\mathbb{R})\to \mathbb{R}$ with $f(AB)=f(BA)$, would be closed under the Poisson bracket ?

Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those functions $f:M_{2n}(\mathbb{R})\to \mathbb{R}$ with $f(AB)=f(BA)$, would be closed under the Poisson bracket ?