$A={A}(x, y, z, t)$, $B={B}(x, y, z, t)$, and $C={C}(x, y, z, t)$, the Nambu bracket is defined by means of the 3-dimensional Jacobian, $$\begin{aligned}
\{{A}, {B}, {C}\}_{\mbox{\tiny{NB}}} 
\equiv \frac{\partial ({A}, {B}, {C})}{\partial (x, y, z)}~.
\label{NB}\end{aligned}$$ In terms of the Nambu bracket, the Nambu equation for any function $f=f(x,y,z)$ can be written as $$\begin{aligned}
\frac{d {f}}{dt} = \{{f}, {H}_1, {H}_2\}_{\mbox{\tiny{NB}}}~.
\label{N-eqf}\end{aligned}$$

It is straightforward to extend the above formalism to a system described by an $N$-plet $x_i$ $(i=1, 2, \cdots, N)$. These variables satisfy the Nambu equations $$\begin{aligned}
\frac{d x_i}{dt} =  \sum_{i_1, \cdots, i_{N-1}=1}^{N}
\varepsilon_{i i_1 \cdots i_{N-1}}
\frac{\partial {H}_1}{\partial x_{i_1}} \cdots 
\frac{\partial {H}_{N-1}}{\partial x_{i_{N-1}}}~,
\label{N-eq-N}\end{aligned}$$ where ${H}_a={H}_a(x_1, x_2, \cdots, x_N)$ $(a=1, \cdots, N-1)$ are $\lq\lq$Hamiltonians" of this system and $\varepsilon_{i i_1 \cdots i_{N-1}}$ is the $N$-dimensional Levi–Civita symbol, the antisymmetric tensor with $\varepsilon_{12 \cdots N} =1$. For any functions ${A}_{\alpha}={A}_{\alpha}(x_1, x_2, \cdots, x_N, t)$ $(\alpha=1, \cdots, N)$, the Nambu bracket is defined by means of the $N$-dimensional Jacobian, $$\begin{aligned}
\{{A}_1, {A}_2, \cdots, {A}_N\}_{\mbox{\tiny{NB}}} 
&\equiv& 
\frac{\partial ({A}_1, {A}_2, \cdots, {A}_N)}{\partial (x_1, x_2, \cdots, x_N)}
\nonumber \\
&=& \sum_{i_1, i_2, \cdots, i_N=1}^{N} \varepsilon_{i_1 i_2 \cdots i_N}
\frac{\partial {A}_1}{\partial x_{i_1}}\frac{\partial {A}_2}{\partial x_{i_2}}
 \cdots \frac{\partial {A}_{N}}{\partial x_{i_N}}~.
\label{NB-N}\end{aligned}$$ In terms of the 