In Goldstein 3rd edision p. 409, it is stated that $$\partial \mathbf{F}=[\mathbf{F}, \mathbf{L}\cdot\mathbf{n}]=\mathbf{n}\times\mathbf{F}\tag{9.121}$$ if $\mathbf{F}$ is a function of system variables $(q,p)$. (The square bracket [] denotes a Poisson bracket.)
What I wonder is if that relation can be derived solely from the algebraic manipulation without relying on $\partial{F}=d\theta\mathbf{n}\times\mathbf{F}$, or in other words, is the property of any system vector $\mathbf{F}(q,p)$ that can be "obtained" solely from the algebraic property of Poisson bracket plus the canonical commutation relation.
Actually, I tried but in vain as below... \begin{align*} \hat{e}_\alpha n^i\epsilon_{ijk}[F^\alpha,q^jp^k]&=\hat{e}_\alpha n^i\epsilon_{ijk}(q^j[F^\alpha,p^k]+[F^\alpha,q^j]p^k)\\ &=\hat{e}_\alpha n^i\epsilon_{ijk}(q^j\frac{\partial F^\alpha}{\partial q_k}+p^j\frac{\partial F^\alpha}{\partial p_k})\\ &=\hat{e}_\alpha\frac{\partial F^\alpha}{\partial \vec{\eta}}\cdot(\hat{n}\times\vec{\eta})\\ &=?=(\hat{n}\times\vec{F}) \end{align*}
When I first learned this subject, I thought that the above relation (which I tried to derive) is the prerequisite for a system vector, but the intonation of the book sounds to say it could actually be algebraically derived for all vectors that are a function of $(q,p)$. Could somebody please provide a clear answer to this?
