octonion

An octonion is an element of a real eight‑dimensional non‑associative algebra. They form the largest normed division algebra over the real numbers, alongside the reals, complexes, and quaternions, as established by Hurwitz's theorem. An octonion can be written as x = x0 + x1 e1 + x2 e2 + x3 e3 + x4 e4 + x5 e5 + x6 e6 + x7 e7 with real coefficients, where 1 is the multiplicative identity and e1,...,e7 are imaginary units. The product is neither commutative nor associative, but it is alternative: any subalgebra generated by two elements is associative. Conjugation is defined by x* = x0 − x1 e1 − ... − x7 e7, and the norm N(x) = x x* = x0^2 + x1^2 + ... + x7^2 is multiplicative: N(xy) = N(x) N(y). Consequently, every nonzero octonion has a multiplicative inverse.

Octonions can be constructed by the Cayley–Dickson process, doubling quaternions. If a and b are quaternions, the product (a, b) · (c, d) is defined as (ac + d b̄, ā d + b c̄), where overlines denote quaternionic conjugation. This yields an eight‑dimensional algebra with the above basis. Another convenient realization is Zorn’s vector‑matrix form, which encodes the same multiplication.

The automorphism group of the octonions is the exceptional Lie group G2, preserving the product. Octonions arise in geometry and algebra, notably the parallelizability of the 7-sphere S7 and the construction of the exceptional Jordan algebra. Their nonassociativity limits algebraic closure, but as a normed division algebra they appear in various areas of theoretical physics and the study of exceptional structures.