To Örebro University

We write the Lagrangian of the general N = 5 three-dimensional superconformal Chern-Simons theory, based on a basic Lie superalgebra, in terms of our recently introduced N = 5 three-algebras. These include N = 6 and N = 8 three-algebras as special cases. When we impose an antisymmetry condition on the triple product, the supersymmetry automatically enhances, and the N = 5 Lagrangian reduces to that of the well known N = 6 theory, including the ABJM and ABJ models.