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superstring theory has also employed division algebras including octonions, the dimensions three, four, six, and ten (all two more than the corresponding division algebra) being distinguished in permitting relations between vectors and spinors and certain isomorphisms between groups that allow the decomposition of a spinor as a pair of elements of that division algebra: 10.1016/0550-3213(83)90584-9;
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superstring theory has also employed division algebras including octonions, the dimensions three, four, six, and ten (all two more than the corresponding division algebra) being distinguished in permitting relations between vectors and spinors and certain isomorphisms between groups that allow the decomposition of a spinor as a pair of elements of that division algebra: T. Kugo and P. Townsend, Nucl. Phys. B 221, 357 (1983) 10.1016/0550-3213(83)90584- 9
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See Baez in for a striking alternative that places the seven elements and unity at the corners of a cube. Planes through the origin (unity) give subalgebras isomorphic to quaternions, lines through the origin give subalgebras isomorphic to complex numbers, and the origin itself the reals. Thus the Fano plane is the projective plane over the two-element field Z2 and is the smallest possible projective plane.
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