  In this paper we construct two kinds of infinite-dimensional 3-Lie algebras
from a given commutative associative algebra, and show that they are all
canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to
Witt algebras, and then study the regular representations of these 3-Lie
algebras and the natural representations of the inner derivation algebras. In
particular, for the second kind of 3-Lie algebras, we find that their regular
representations are Harish-Chandra modules, and the inner derivation algebras
give rise to intermediate series modules of the Witt algebras and contain the
smallest full toroidal Lie algebras without center.
