Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, meaning it is a hypercomplex number system; Octonions are usually represented by the capital letter O, using boldface O or blackboard bold O {displaystyle mathbb {O} } . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.The octonions were discovered in 1843 by John T. Graves, inspired by his friend W. R. Hamilton's discovery of quaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December 1843. He first published his result slightly later than Arthur Cayley's article. The octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery.The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions:Octonionic multiplication is neither commutative:The octonions play a significant role in the classification and construction of other mathematical entities. For example, the exceptional Lie group G2 is the automorphism group of the octonions, and the other exceptional Lie groups F4, E6, E7 and E8 can be understood as the isometries of certain projective planes defined using the octonions. The set of self-adjoint 3 × 3 {displaystyle 3 imes 3}   octonionic matrices, equipped with a symmetrized matrix product, defines the Albert algebra. In discrete mathematics, the octonions provide an elementary derivation of the Leech lattice, and thus they are closely related to the sporadic simple groups.There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly 7 maximal orders containing it. These 7 maximal orders are all equivalent under automorphisms. The phrase 'integral octonions' usually refers to a fixed choice of one of these seven orders.