In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is...

mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented...

mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero...

Dickson. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples...

called quaternion algebras.  They are associative but not commutative. Composition algebras of dimension 8 are called octonion algebras.  They are neither...

in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. Alternative...

compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen...

trivial involution, and quadratic form z2. The algebra of bioctonions is an example of an octonion algebra. For any pair of bioctonions y and z, N ( y z...

In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras...

rings. An algebra is unital or unitary if it has an identity element e with ex = x = xe for all x in the algebra. For example, the octonions are unital...

of B. Composition algebra Cyclic algebra Octonion algebra Hurwitz quaternion order Hurwitz quaternion See Pierce. Associative algebras. Springer. Lemma...

non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there...

algebra Kac–Moody algebra Kleene algebra Leibniz algebra Lie algebra Lie superalgebra Malcev algebra Matrix algebra Non-associative algebra Octonion algebra...

octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution. Over any algebraically closed...

composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method was formalized by...

Octonions Lie algebras Jordan algebras Alternative algebras Flexible algebras Power-associative algebras The definition of an associative K-algebra with...

{H} } , and the octonions O {\displaystyle \mathbb {O} } , and the Frobenius theorem says the only real associative division algebras are R {\displaystyle...

ISBN 0-387-95447-3. MR 2014924. Okubo, Susumo (1995). Introduction to Octonion and Other Non-Associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical...

split-octonions. More generally, the set of invertible elements in any octonion algebra over a field F forms a Moufang loop, as does the subset of unit norm...

mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of...

extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The next extension gives the...

matrices Generalized Clifford algebra Geometric algebra Higher-dimensional gamma matrices Hypercomplex number Octonion Paravector Quaternion Spin group...

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...

if xyz = 1 then a(x)b(y)c(z) = 1. For alternative division algebras such as the octonions the two definitions of isotopy are equivalent, but in general...

exceptional Jordan algebra (it is exceptional because the octonions are not associative). This was the first example of an Albert algebra. Its automorphism...

Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M2) and Tr(M3) of the hermitian octonion matrix: M...

ISBN 0-8176-3345-6, MR 0885089 Okubo, Susumu (1995), Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical...

Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification...

Dickson. For example, the group G2 is the automorphism group of an octonion algebra over k. By contrast, the Chevalley groups of type F4, E7, E8 over a...

Sbitneva; I. P. Shestakov (2006). "§17.2 Octonion algebra and its regular bimodule representation". Non-associative algebra and its applications. CRC Press. p...