  The fermionic fields of one generation of the Standard Model, including the
Lorentz spinor degrees of freedom, can be identified with components of a
single real 64-dimensional semi-spinor representation S of the group
Spin(11,3). We describe an octonionic model for Spin(11,3) in which the
semi-spinor representation gets identified with S=OxO', where O,O' are the
usual and split octonions respectively. It is then well-known that choosing a
unit imaginary octonion u in Im(O) equips O with a complex structure J.
Similarly, choosing a unit imaginary split octonion u' in Im(O') equips O' with
a complex structure J', except that there are now two inequivalent complex
structures, one parametrised by a choice of a timelike and the other of a
spacelike unit u'. In either case, the identification S=OxO' implies that there
are two natural commuting complex structures J, J' on S. Our main new
observation is that the subgroup of Spin(11,3) that commutes with both J, J' on
S is the direct product Spin(6) x Spin(4) x Spin(1,3) of the Pati-Salam and
Lorentz groups, when u' is chosen to be timelike. The splitting of S into
eigenspaces of J corresponds to splitting into particles and anti-particles.
The splitting of S into eigenspaces of J' corresponds to splitting of Lorentz
Dirac spinors into two different chiralities. We also study the simplest
possible symmetry breaking scenario with the "Higgs" field taking values in the
representation that corresponds to 3-forms in R^{11,3}. We show that this Higgs
can be designed to transform as the bi-doublet of the left/right symmetric
extension of the SM, and thus breaks Spin(11,3) down to the product of the SM,
Lorentz and U(1)_{B-L} groups, with the last one remaining unbroken. This
3-form Higgs field also produces the Dirac mass terms for all the particles.
