  We have recently proposed a Lagrangian in trace dynamics at the Planck scale,
for unification of gravitation, Yang-Mills fields, and fermions. Dynamical
variables are described by odd-grade (fermionic) and even-grade (bosonic)
Grassmann matrices. Evolution takes place in Connes time. At energies much
lower than Planck scale, trace dynamics reduces to quantum field theory. In the
present paper we explain that the correct understanding of spin requires us to
formulate the theory in 8-D octonionic space. The automorphisms of the octonion
algebra, which belong to the smallest exceptional Lie group $G_2$, replace
space-time diffeomorphisms and internal gauge transformations, bringing them
under a common unified fold. Building on earlier work by other researchers on
division algebras, we propose the Lorentz-weak unification at the Planck scale,
the symmetry group being the stabiliser group of the quaternions inside the
octonions. This is one of the two maximal subgroups of $G_2$, the other one
being $SU(3)$, the element preserver group of octonions. This latter group,
coupled with $U(1)_{em}$, describes the electro-colour symmetry, as shown
earlier by Furey. We predict a new massless spin one boson [the Lorentz boson]
which should be looked for in experiments. Our Lagrangian correctly describes
three fermion generations, through three copies of the group $G_2$, embedded in
the exceptional Lie group $F_4$. This is the unification group for the four
fundamental interactions, and it also happens to be the automorphism group of
the exceptional Jordan algebra. Gravitation is shown to be an emergent
classical phenomenon. Whereas at the Planck scale, there is present a quantised
version of the Lorentz symmetry, mediated by the Lorentz boson. We argue that
at sub-Planck scales, the self-adjoint part of the octonionic trace dynamics
bears a relationship with string theory in eleven dimensions.
