Triality principle and G₂ group in spinor language

The presented paper is aimed at providing a systematic study of a relation between octonions and spinors corresponding to S⁷ 7‐sphere, starting from a natural point of view, enabling us to endow spinor space S₊(8,0) with octonion algebra structures. As a result we arrive at formulations of triality principle in its finite form in terms of vector and fundamental representations of Spin (8) group—both for spinors and vectors. The group of automorphisms of octonion algebra, as well as its Lie algebra, gains clear interpretation in the context. The method proposed is purely algebraic and could be applied as well to 𝒞(4,4) Clifford algebra corresponding to ₄S⁷ indefinite 7‐sphere geometry.