Nambu bracket is a natural generalization of Poisson bracket. A very distinguished property is its decomposability. This is investigated from the second order term of the fundamental identity. In this paper, we shall study the first order term of the fundamental identity and get a relation with the Schouten-Nijenhuis bracket. And also we shall show that for a given Poisson structure, the top power of it gives a Nambu structure. We shall characterize the Godbillon-Vey class of the foliation defined from a regular Nambu tensor.