  Motivated by the recent proposal of an N=8 supersymmetric action for multiple
M2-branes, we study the Lie 3-algebra in detail. In particular, we focus on the
fundamental identity and the relation with Nambu-Poisson bracket. Some new
algebras not known in the literature are found. Next we consider cubic matrix
representations of Lie 3-algebras. We show how to obtain higher dimensional
representations by tensor products for a generic 3-algebra. A criterion of
reducibility is presented. We also discuss the application of Lie 3-algebra to
the membrane physics, including the Basu-Harvey equation and the Bagger-Lambert
model.
