The Split Octonions

This chapter explores the split-octonion arithmetic. We begin with a non-optical derivation of the Lorentz transformations, build the split-octonions, show how this breaks if we try to go to four spatial dimensions and explore the arithmetic of this algebra over lattice structures.

This presentation is intended for people with a math background through linear algebra, with familiarity with the complex numbers. Familiarity with Clifford algebras will be helpful, but not necessary. The split-octonions are closely related to this algebra and the notation here will be borrowed from Clifford algebras.

The algebra of the complex numbers will be used to introduce the new notation. Properties from abstract algebra will be used as justification of certain statements that advanced students might question.

Section 1: Complex Numbers: Planar Rotations
Section 2: Split-Complex Numbers: Lorentz Transformations
Section 3: Cayley-Dickson Construction
Section 4: Quaternions: 3D rotations
Section 5: Split-Quaternions: Thomas Rotation

Note: The algebra originally used here is not a composition algebra, and is thus not the Split-Octonions. This paradigm may be revisited at some future time. The multiplication table has been edited to reflect the actual split-octonion algebra.
Section 6: Split-Octonions: Helicity
Section 7: Split-Sedonions?
Section 8: Split-Octonion Lattices

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