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 $\mathfrak{L}\mathrm{/}\mathbb{F}{L}_{0,0}^0$ 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-$W_{\mathrm{\infty }}$ algebra can be embedded in $\mathfrak{L}$.