This article is a write-up of the talk given in one of the mini-symposia of the 2024 European Congress of Mathematicians. I will explain some basics of the representation theory underlying Spin(10) and SU(5) Grand Unified Theories. I will also explain the characterisation of the Standard Model gauge group \(G_\textrm{SM}\) as a subgroup of Spin(10) that was developed in Krasnov, J Math Phys 65(8):082302 (2024), [1]. Thus, the symmetry breaking required to obtain \(G_\textrm{SM}\subset \textrm{Spin}(10)\) can be seen to rely on two suitably aligned commuting complex structures on \({\mathbb R}^{10}\) . The required complex structures can in turn be encoded in a pair of pure spinors of \(\textrm{Spin}(10)\) . The condition that the complex structures are commuting and suitably aligned translates into the requirement that the respective pure spinors are orthogonal and that their sum is again a pure spinor. The most efficient description of spinors, and in particular pure spinors of Spin(10) is via the octonionic model of the latter, and this is how octonions enter the story.