Octonions - Properties

Properties

Octonionic multiplication is neither commutative:

if

nor associative:

if

The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of O is isomorphic to R, C, or H, all of which are associative. Because of their non-associativity, octonions don't have matrix representations, unlike quaternions.

The octonions do retain one important property shared by R, C, and H: the norm on O satisfies

This implies that the octonions form a nonassociative normed division algebra. The higher-dimensional algebras defined by the Cayley–Dickson construction (e.g. the sedenions) all fail to satisfy this property. They all have zero divisors.

Wider number systems exist which have a multiplicative modulus (e.g. 16 dimensional conic sedenions). Their modulus is defined differently from their norm, and they also contain zero divisors.

It turns out that the only normed division algebras over the reals are R, C, H, and O. These four algebras also form the only alternative, finite-dimensional division algebras over the reals (up to isomorphism).

Not being associative, the nonzero elements of O do not form a group. They do, however, form a loop, indeed a Moufang loop.