  We provide a new characterisation of the Standard Model gauge group GSM as a
subgroup of Spin(10). The new description of GSM relies on the geometry of pure
spinors. We show that GSM is the subgroup that stabilises a pure spinor Psi_1
and projectively stabilises another pure spinor Psi_2, with Psi_1, Psi_2
orthogonal and such that their arbitrary linear combination is still a pure
spinor. Our characterisation of GSM relies on the facts that projective pure
spinors describe complex structures on R^{10}, and the product of two commuting
complex structures is a what is known as a product structure. For the pure
spinors Psi_1, Psi_2 satisfying the stated conditions the complex structures
determined by Psi_1, Psi_2 commute and the arising product structure is R^{10}
= R^6 + R^4, giving rise to a copy of Pati-Salam gauge group inside Spin(10).
Our main statement then follows from the fact that GSM is the intersection of
the Georgi-Glashow SU(5) that stabilises Psi_1, and the Pati-Salam Spin(6) x
Spin(4) arising from the product structure determined by Psi_1, Psi_2. We have
tried to make the paper self-contained and provided a detailed description of
the creation/annihilation operator construction of the Clifford algebras Cl(2n)
and the geometry of pure spinors in dimensions up to and including ten.
