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 [S. Cherkis, C. Saemann, arXiv: 0807.0808 [hep-th]], this gives rise to a three-dimensional N=2 Chern-Simons-matter theory with exceptional gauge group G2 and with global symmetry SU(4)×U(1). This gives a possible candidate for the theory on multiple M2-branes with G2 gauge symmetry.