  A recent series of works by M. Dubois-Violette, I. Todorov and S. Drenska
characterised the SM gauge group GSM as the subgroup of SO(9) that, in the
octonionic model of the later, preserves the split O=C+C3 of the space of
octonions into a copy of the complex plane plus the rest. This description,
however, proceeded via the exceptional Jordan algebras J3(O), J2(O) and and
this sense remained indirect. One of the goals of this paper is to provide as
explicit description as possible and also clarify the underlying geometry. The
other goal is to emphasise the role played by different complex structures in
the spaces O and O2. We provide a new characterisation of GSM: The group GSM is
the subgroup of Spin(9) that commutes with of a certain complex structure J in
the space O2 of Spin(9) spinors. The complex structure J is parametrised by a
choice of a unit imaginary octonion. This characterisation of GSM is
essentially octonionic in the sense that J is restrictive because octonions are
non-associative. The quaternionic analog of J is the complex structure in the
space H2 of Spin(5) spinors that commutes with all Spin(5) transformations.
