Does anybody know how to show the Jacobi identity for the Nambu bracket in $\mathbb{R}^3$? The Nambu bracket with respect to $c \in \mathcal{F}(\mathbb{R}^3)$ is defined as $\{F,G\}_c = \langle\nabla c , \nabla F \times\nabla G\rangle $ where $F,G \in \mathcal{F}(\mathbb{R}^3).$
I don't know if this will help but, if one shows that the homomorphism $ F \rightarrow \{F, \bullet\}_c = \langle\nabla \bullet , \nabla c \times \nabla F\rangle,$ between $\mathcal{F}(\mathbb{R}^3)$ and the divergenceless vector fields $\nabla c \times \nabla F,$ preserve the lie algebra structure then it is done...
Thank you very much for the help!