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 $mathfrak{L}$.