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

example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form...

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

origin are proportional to j exp(aj). In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson...

Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in...

hi is a null vector. The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light...

Moufang loop. The nonzero octonions form a nonassociative Moufang loop under octonion multiplication. The subset of unit norm octonions (forming a 7-sphere...

defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution...

Pseudoquaternions (Yaglom 1968 Rosenfeld 1988) Pauli matrices Split-biquaternions Split-octonions Dual quaternions Karzel, Helmut & Günter Kist (1985) "Kinematic Algebras...

systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real...

include split-quaternions (coquaternions), split-octonions, biquaternions C ⊗ H {\displaystyle \mathbb {C} \otimes \mathbb {H} } , and complex octonions C ⊗...

the direct sum of two quaternion algebras – van der Waerden 1985 Split-octonions Clifford, W. K. (1873). "Preliminary Sketch of Biquaternions". In Tucker...

quadratic form x2 − y2, quaternions and split-quaternions, octonions and split-octonions. Every composition algebra has an associated bilinear form B(x...

5-manifolds: Here G = Aut(Os) is the automorphism group of the algebra Os of split octonions, a closed subgroup of SO(3,4), and P is the intersection of G with...

exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical...

dem Mathematischen Seminar der Universität Hamburg. Zorn showed that split-octonions could be represented by a mixed-style of matrices called Zorn's vector-matrix...

The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting...

largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The...

The 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...

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...

ramifies is called the discriminant of B. Composition algebra Cyclic algebra Octonion algebra Hurwitz quaternion order Hurwitz quaternion See Pierce. Associative...

ISBN 978-0-486-66341-8. Lasenby, A. (2022), Some recent results for SU(3) and Octonions within the Geometric Algebra approach to the fundamental forces of nature...

generation of 16 fermions can be put into the form of an octonion with each element of the octonion being an 8-vector. If the 3 generations are then put in...

a p-adic field. Image: Algebraic group: the automorphism group of a split octonion algebra. Special fields: Exists over a finite field, the reals, a p-adic...

E. 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...

sedenions are obtained by applying the Cayley–Dickson construction to the octonions, which can be mathematically expressed as S = C D ( O , 1 ) {\displaystyle...

O} }   stand for the real numbers, complex numbers, quaternions, and octonions. In the symbols such as E6−26 for the exceptional groups, the exponent...

type G2(q) is usually constructed as the automorphism groups of the split octonions. Hence, it has a natural representation as a subgroup of the 7-dimensional...

for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued...

using an algebra that is the tensor product of the quaternions and the octonions, and is also known as a Rosenfeld projective plane, though it does not...