\frac{d \tilde{f}}{dt} = \frac{\partial(f, H)}{\partial(q, p)} 
= \frac{\partial(\tilde{f}, \tilde{H})}{\partial(x, y)} \{x, y\}_{\mbox{\tiny{PB}}}
+ \frac{\partial(\tilde{f}, \tilde{H})}{\partial(y, z)} \{y, z\}_{\mbox{\tiny{PB}}}
+ \frac{\partial(\tilde{f}, \tilde{H})}{\partial(z, x)} \{z, x\}_{\mbox{\tiny{PB}}}~,
\label{H-eq(N=3)}\end{aligned}$$ where $\tilde{H}(x, y, z) = H(q, p)$. Note that $q$, $p$, and $H$ are, in general, not uniquely determined as functions of $x$, $y$, and $z$.

Introducing a function $\tilde{G}=\tilde{G}(x, y, z)$ that satisfies the conditions $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x} = \frac{\partial(y, z)}{\partial(q, p)}~,~~
\frac{\partial \tilde{G}}{\partial y} = \frac{\partial(z, x)}{\partial(q, p)}~,~~
\frac{\partial \tilde{G}}{\partial z} = \frac{\partial(x, y)}{\partial(q, p)}~,
\label{xyzG}\end{aligned}$$ Eq. (\[H-eq(N=3)\]) is rewritten as the Nambu equation in the form of Eq. (\[N-eqf\]), $$\begin{aligned}
\frac{d \tilde{f}}{dt} = \{\tilde{f}, \tilde{H}, \tilde{G}\}_{\mbox{\tiny{NB}}}~,
\label{H-eqf(N=3)}\end{aligned}$$ where we use the formula $$\begin{aligned}
\frac{\partial (\tilde{A}, \tilde{B}, \tilde{C})}{\partial (x, y, z)}
= \frac{\partial (\tilde{A}, \tilde{B})}{\partial (x, y)}
\frac{\partial \tilde{C}}{\partial z}
+ \frac{\partial (\tilde{A}, \tilde{B})}{\partial (y, z)}
\frac{\partial \tilde{C}}{\partial x}
+ \frac{\partial (\tilde{A}, \tilde{B})}{\partial (z, x)}
\frac{\partial \tilde{C}}{\partial y}~.
\label{J(N=3)}\end{aligned}$$ The conditions (\[xyzG\]) are compactly expressed as $$\begin{aligned}
\frac{\partial \tilde{G}}{\partial x_i} 
= \frac{1}{2} \sum_{j, k=1}^3 \varepsilon_{ijk}\{x_j, x_k\}_{\mbox{\tiny{PB}}}~~~
\mbox{or}~~~
\sum_{k=1}^3 \varepsilon_{ijk} \frac{\partial \tilde{G}}{\partial x_k} 
= \{x_i, x_j\}_{\mbox{\tiny{PB}}}~.
\label{xyzG2}\end{aligned}$$ In Appendix A, the Nambu equations in the form of Eq. (\[N-eq\]) are also derived from a Hamiltonian system with a canonical doublet $(q, p)$ using 