Math 2.0

Nonassociative rings and algebras Composition algebras and the Cayley–Dickson construction Object

Split-octonions

Builds on Cayley–Dickson constructionThe quaternionsComposition algebraOctonions

Description

The split-octonions form the 88-dimensional real algebra CD(ℍ,1)\mathrm{CD}(\mathbb{H}, 1), the Cayley–Dickson double of the quaternions ℍ\mathbb{H} with parameter μ=1\mu = 1. A split-octonion is a pair (a,b)(a, b) of quaternions, with the product (a,b)(c,d)=(ac+dˉb,  da+bcˉ).(a, b)(c, d) = (ac + \bar{d} b,\; da + b\bar{c}). The unit is (1,0)(1, 0), the conjugate of (a,b)(a, b) is (aˉ,−b)(\bar{a}, -b), and the norm is N(a,b)=∣a∣2−∣b∣2N(a, b) = \abs{a}^2 - \abs{b}^2.

Remarks

The norm is multiplicative and has signature (4,4)(4, 4), so the split-octonions are a composition algebra and an octonion algebra over ℝ\R. They are alternative and not associative. They have zero divisors: with ℓ=(0,1)\ell = (0, 1), one has ℓ2=(1,0)=1\ell^2 = (1, 0) = 1, so (1+ℓ)(1−ℓ)=1−ℓ2=0(1 + \ell)(1 - \ell) = 1 - \ell^2 = 0. Up to isomorphism, the octonion algebras over ℝ\R are the split-octonions and the octonions 𝕆\mathbb{O}.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Cayley–Dickson construction, concept, no evidence levelCayley–Dickson constr…conceptComposition algebra, concept, no evidence levelComposition algebraconceptGroup, concept, no evidence levelGroupconceptNonassociative ring, concept, no evidence levelNonassociative ringconceptRing, definition, no evidence levelRingdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleOctonions, object, no evidence levelOctonionsobjectSplit-octonions, object, no evidence levelSplit-octonionsobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cayley–Dickson construction concept
  2. Composition algebra concept
  3. Group concept
  4. Nonassociative ring concept
  5. Ring definition
  6. Unit of a ring definition
  7. Division ring definition
  8. The quaternions example
  9. Octonions object
  10. Split-octonions object

Further reading

  1. Web https://en.wikipedia.org/wiki/Split-octonion
About this page
TagV17Y6PNA13
URIm2://V17Y6PNA13
Kindobject
WritingW0 stub
MSC17A75, 17D05
Instance ofOctonion algebra
UsesCayley–Dickson construction, The quaternions, Composition algebra, Octonions
Same as
Changed3 November 2026

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