The infinite dimensional Unital 3-Lie Poisson algebra

From a commutative associative algebra $A$, the infinite dimensional unital 3-Lie Poisson algebra~$\mathfrak{L}$~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $\mathfrak{L}$ is discussed. It is proved that: (1) there is a minimal set of generators $S$ consisting of six vectors; (2) the quotient algebra $\mathfrak{L}/\mathbb{F}L_{0, 0}^0$ is a simple 3-Lie Poisson algebra; (3) four important infinite dimensional 3-Lie algebras: 3-Virasoro-Witt algebra $\mathcal{W}_3$, $A_\omega^\delta$, $A_{\omega}$ and the 3-$W_{\infty}$ algebra can be embedded in $\mathfrak{L}$.