Three-algebras and $\mathcal{N}=6$ Chern-Simons gauge theories

We derive the general form for a three-dimensional scale-invariant field theory with $\mathcal{N}=6$ supersymmetry, $SU(4)$ R-symmetry and a $U(1)$ global symmetry. The results can be written in terms of a 3-algebra in which the triple product is not antisymmetric. For a specific choice of 3-algebra we obtain the $\mathcal{N}=6$ theories that have been recently proposed as models for M2-branes in an ${\mathbb{R}}^8/{\mathbb{Z}}_k$ orbifold background.