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

Sedenions

Builds on Cayley–Dickson constructionOctonionsAlternative algebraDivision algebra

Description

The sedenions 𝕊\mathbb{S} are the 1616-dimensional real algebra obtained from the octonions 𝕆\mathbb{O} by the Cayley–Dickson construction with parameter μ=−1\mu = -1. So a sedenion is a pair (a,b)(a, b) of octonions, with the product (a,b)(c,d)=(ac−dˉb,  da+bcˉ),(a, b)(c, d) = (ac - \bar{d} b,\; da + b\bar{c}), where xˉ\bar{x} is the octonion conjugate. The unit is (1,0)(1, 0), and the conjugate of (a,b)(a, b) is (aˉ,−b)(\bar{a}, -b).

Remarks

The sedenions are not commutative, not associative and not alternative. They are still flexible and power-associative, as are all algebras of the Cayley–Dickson sequence that starts from ℝ\R. They are not a division algebra: by the Bott–Milnor–Kervaire theorem a real division algebra has dimension 11, 22, 44 or 88. As 𝕊\mathbb{S} has finite dimension, it therefore has zero divisors: nonzero x,yx, y with xy=0xy = 0. For the same reason the norm N(a,b)=N(a)+N(b)N(a, b) = N(a) + N(b) is not multiplicative, so 𝕊\mathbb{S} is not a composition algebra.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Alternative algebra, concept, no evidence levelAlternative algebraconceptCayley–Dickson construction, concept, no evidence levelCayley–Dickson constr…conceptDivision algebra, concept, no evidence levelDivision 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 levelOctonionsobjectSedenions, object, no evidence levelSedenionsobject
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. Alternative algebra concept
  2. Cayley–Dickson construction concept
  3. Division algebra concept
  4. Group concept
  5. Nonassociative ring concept
  6. Ring definition
  7. Unit of a ring definition
  8. Division ring definition
  9. The quaternions example
  10. Octonions object
  11. Sedenions object

Further reading

  1. Web https://en.wikipedia.org/wiki/Sedenion
About this page
TagYADT1TBZ5A
URIm2://YADT1TBZ5A
Kindobject
WritingW0 stub
MSC17A99, 17A45
UsesCayley–Dickson construction, Octonions, Alternative algebra, Division algebra
Same as
Changed3 November 2026

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