Octonion
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The octonions are the non-associative extension of the quaternions. They were discovered by John T. Graves in 1843, and independently by Arthur Cayley, who first published them in 1845. They are sometimes called Cayley numbers.
Octonions form an 8-dimensional algebra over real numbers and can be understood as an ordered octet of real numbers. Each octonion forms a linear combination of the base: 1, e1, e2, e 3, e4, e5, e6, e7. The way to multiply octonions is given in the following table:
| · | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| 1 | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| e1 | e1 | -1 | e4 | e7 | - Hey.2 | e6 | - Hey.5 | - Hey.3 |
| e2 | e2 | - Hey.4 | -1 | e5 | e1 | - Hey.3 | e7 | - Hey.6 |
| e3 | e3 | - Hey.7 | - Hey.5 | -1 | e6 | e2 | - Hey.4 | e1 |
| e4 | e4 | e2 | - Hey.1 | - Hey.6 | -1 | e7 | e3 | - Hey.5 |
| e5 | e5 | - Hey.6 | e3 | - Hey.2 | - Hey.7 | -1 | e1 | e4 |
| e6 | e6 | e5 | - Hey.7 | e4 | - Hey.3 | - Hey.1 | -1 | e2 |
| e7 | e7 | e3 | e6 | - Hey.1 | e5 | - Hey.4 | - Hey.2 | -1 |
This product is neither commutative nor associative. Because of this non-associativity, octonions, unlike quaternions, do not admit a matrix representation.
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