### Abstract

We develop the basic theory of Octonions, including various identities and properties of the octonions and of the octonionic product, a description of 7D isometries and representations of orthogonal transformations. To this end we first develop the theory of the vector cross product in 7 dimensions. The development of the theory of Octonions is inspired by that of the theory of Quaternions by Lawrence Paulson. However, we do not work within the type class real_algebra_1 because the octonionic product is not associative.

### License

BSD License

### Topics

- Mathematics/Algebra
- Mathematics/Geometry

### Session Octonions

- Cross_Product_7
- Octonions

## Cite

×

Octonions-AFP

    @article{Octonions-AFP,
      author  = {Angeliki Koutsoukou-Argyraki},
      title   = {Octonions},
      journal = {Archive of Formal Proofs},
      month   = {September},
      year    = {2018},
      note    = {\url{https://isa-afp.org/entries/Octonions.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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