Math 2.0

Nonassociative rings and algebras Further nonassociative algebras Object

Coxeter's integral octonions

Builds on OctonionsOctonion algebraMaximal orderE8 latticeMoufang loopHurwitz quaternions

Description

Let 𝕆\mathbb{O} be the octonions, with standard basis 1,e1,…,e71, e_1, \ldots, e_7, conjugation x↦xˉx \mapsto \bar{x} and norm N(x)=xxˉN(x) = x\bar{x}. The Gravesian octonions are the octonions whose eight coordinates are all integers. They form a subring of 𝕆\mathbb{O} that is a lattice of rank 88, on which NN and the trace x+xˉx + \bar{x} take integer values. But it is not maximal among the subrings with these properties. There are exactly seven maximal subrings of this kind that contain the Gravesian octonions. Coxeter's integral octonions are any one of these seven rings. The automorphisms of 𝕆\mathbb{O} permute the seven rings transitively, so the choice of one of them is harmless. Coxeter studied them in his 1946 paper on integral Cayley numbers.

Remarks

They are the octonion analogue of the Hurwitz quaternions, a maximal order in the quaternions. Besides the integer points they contain points with half-integer coordinates. As a lattice in 𝕆≅ℝ8\mathbb{O} \cong \R^8 with the quadratic form 2N2N, they form a copy of the E8 lattice. So the integral octonions of norm 11 are 240240 elements, matching the 240240 roots of E8E_8. These 240240 units are closed under multiplication and inverses and form a Moufang loop that is not a group, because octonion multiplication is not associative.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectEuler's four-square identity, statement, no evidence levelEuler's four-square i…statementGroup, concept, no evidence levelGroupconceptLattice (group), concept, no evidence levelLattice (group)conceptMaximal order, concept, no evidence levelMaximal orderconceptMoufang loop, concept, no evidence levelMoufang loopconceptNonassociative ring, concept, no evidence levelNonassociative ringconceptOctonion algebra, concept, no evidence levelOctonion algebraconceptOrder (ring theory), concept, no evidence levelOrder (ring theory)conceptQuaternion algebra, concept, no evidence levelQuaternion algebraconceptLagrange's four-square theorem, statement, no evidence levelLagrange's four-squar…statementE8 lattice, object, no evidence levelE8 latticeobjectRing, 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 levelOctonionsobjectHurwitz quaternions, object, no evidence levelHurwitz quaternionsobjectCoxeter's integral octonions, object, no evidence levelCoxeter's integral oc…object
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. Euclidean space object
  2. Euler's four-square identity statement
  3. Group concept
  4. Lattice (group) concept
  5. Maximal order concept
  6. Moufang loop concept
  7. Nonassociative ring concept
  8. Octonion algebra concept
  9. Order (ring theory) concept
  10. Quaternion algebra concept
  11. Lagrange's four-square theorem statement
  12. E8 lattice object
  13. Ring definition
  14. Unit of a ring definition
  15. Division ring definition
  16. The quaternions example
  17. Octonions object
  18. Hurwitz quaternions object
  19. Coxeter's integral octonions object

Further reading

  1. Web https://en.wikipedia.org/wiki/Octonion, Integral octonions
About this page
TagE8WYXVENC6
URIm2://E8WYXVENC6
Kindobject
WritingW0 stub
MSC17A75, 11H06, 17D05
UsesOctonions, Octonion algebra, Maximal order, E8 lattice, Moufang loop, Hurwitz quaternions
Changed3 November 2026

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