I have trouble to understand the definition of the rigid body poisson brackets. In the book of Marsden and Ratiu "Introduction to Mechanics and Symmetry", Chapter 10.1, they introduce the Poisson bracket as follows: Let $\Pi \in \mathbb{R}^3 \simeq \mathfrak{so(3)}$, the Lie algebra of $\mathrm{SO}(3)$. The gradients of the functions $F,G \in C^\infty((\mathbb{R}^3)^*,\mathbb{R})$ are evaluated at the point $\Pi$. The bracket has form $$\{F,G\}=-\langle \Pi, \nabla F(\Pi) \times \nabla G (\Pi)\rangle$$ A few pages later, they introduce a Casimir function w.r.t. the rigid body Poisson bracket: $$C(\Pi)=\frac12 (\Pi_1^2+\Pi_2^2+\Pi_3^2).$$ This function fullfills the property that $$\{C,G\}= -\langle \Pi, \nabla C(\Pi) \times \nabla G (\Pi)\rangle=- \langle \Pi, \Pi \times \nabla G (\Pi)\rangle=-\langle \nabla G,\Pi \times \Pi\rangle =0$$ but this means the Cartesian coordinates are $ \Pi_1, \Pi_2, \Pi_3$, because the gradient is a derivation w.r.t. the Cartesian coordinates. But this means $\Pi$ must have the form $\Pi=(\Pi_1,\Pi_2,\Pi_3)^T$, otherwise the argumentation of the Casimir function wouldn´t work. But this means $\Pi$ isn't an arbitrary point in $\mathbb{R}^3$, but later $\Pi$ is a position vector of a point on an arbitrary sphere.
I hope you understand my problem. I would be glad if someone could help me.