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 e1e2e3e4e5e6e7
1 1e1e2e3e4e5e6e7
e1e1-1 e4e7- Hey.2e6- Hey.5- Hey.3
e2e2- Hey.4-1 e5e1- Hey.3e7- Hey.6
e3e3- Hey.7- Hey.5-1e6e2- Hey.4e1
e4e4e2- Hey.1- Hey.6-1 e7e3- Hey.5
e5e5- Hey.6e3- Hey.2- Hey.7-1 e1e4
e6e6e5- Hey.7e4- Hey.3- Hey.1-1e2
e7e7e3e6- Hey.1e5- 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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