Octonions, G₂ and generalized Lie 3-algebras

We construct an explicit example of a generalized Lie 3-algebra from the octonions. In combination with the result of arXiv:0807.0808, this gives rise to a three-dimensional N=2 Chern-Simons-matter theory with exceptional gauge group G_2 and with global symmetry SU(4)\times U(1). This gives a possible candidate for the theory on multiple M2-branes with G_2 gauge symmetry.