Octonionic Selfduality for SuperMembranes

In this work we study the recently introduced octonionic duality for membranes. Restricting the self - duality equations to seven space dimensions, we provide various forms for them which exhibit the symmetries of the octonionic and quaternionic structure. These forms may turn to be useful for the question of the integrability of this system. Introducing a consistent quadratic Poisson algebra of functions on the membrane we are able to factorize the time dependence of the self - duality equations. We further give the general linear embeddings of the three dimensional system into the seven dimensional one using the invariance of the self-duality equations under the exceptional group G_2.