<!-- Generated from https://www.teclamairme.com/#group-T23 by readviews.py; never edited by hand. -->

# Glossary cards — Chapter ᚖ(W): The third landing

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T23, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch23.md)

*Octonions 𝕆: eight real components, the loss of associativity, the alternative law, a Moufang identity, and the Fano plane as a multiplication table. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Ór — `ᚖ` {#g-og-or}

*Ogham letter*

A supplementary letter whose name means ‘gold’, from Latin aurum.

Ór means ‘gold’, borrowed from Latin aurum, and is the second of the five forfeda. Only two of the three word-ogham lists have an entry for it: sruithem aicde, “most venerable substance” and lí crotha, “splendour of form”; the Cú Chulainn list is silent. The Auraicept glosses feorus no edind, spindle tree or ivy. It appears twice in the inscribed corpus, at Killogrone in County Kerry and on a cross slab at Formaston in Aberdeenshire—which is to say, once in Ireland and once in Pictland. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.18](ch02-refs.md#r-2.18), [2.19](ch02-refs.md#r-2.19)

### an associator with a repeated entry — `(e₁e₁)e₂ − e₁(e₁e₂)` {#g-assoc0}

*Integer or rational*

0 by the alternative law, a surviving associativity promise the octonions keep.

The associator [a, b, c] = (ab)c − a(bc) measures the failure of associativity. An algebra is alternative when the associator vanishes whenever two of its three arguments coincide, so that (aa)b = a(ab) and (ba)a = b(aa); the octonions, found by John Graves in December 1843 and independently by Arthur Cayley, who printed them in 1845, are alternative, so (e₁e₁)e₂ − e₁(e₁e₂) is exactly 0. Artin’s theorem goes further: any two octonions generate an associative subalgebra, which is why octonion powers are unambiguous. A real number computed in 𝕆, hence the integer color.

Cites: [23.1](ch23-refs.md#r-23.1), [23.2](ch23-refs.md#r-23.2), [23.3](ch23-refs.md#r-23.3)

See also: [the octonions](ch23-cards.md#g-octonions)

### an associator of three independent units — `(e₁e₂)e₄ − e₁(e₂e₄)` {#g-assoc}

*Non-real object*

Nonzero, equal to 2e₇ under the Cayley–Dickson table: octonions do not associate.

With three independent imaginary units the associator does not vanish: under the Cayley–Dickson multiplication table, (e₁e₂)e₄ − e₁(e₂e₄) = 2e₇, verified by computation. It is a nonzero octonion, tinted as a non-real object, and it has an inverse because the octonions are a division algebra: the norm is multiplicative, so every nonzero octonion has an inverse (Graves, 1843), and 0 divided by it is 0. Hurwitz’s theorem of 1898 says only that there are no further algebras of the kind. Under the Fano-plane labeling in Baez’s 2002 survey the triple e₁, e₂, e₄ is a quaternion line and a nonzero associator comes instead from e₁, e₂, e₃: (e₁e₂)e₃ − e₁(e₂e₃) = −2e₆; the book uses the Cayley–Dickson table so that this chapter and Chapter ᚗ(V) agree.

Cites: [23.19](ch23-refs.md#r-23.19), [23.4](ch23-refs.md#r-23.4), [23.6](ch23-refs.md#r-23.6), [23.7](ch23-refs.md#r-23.7)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [Hurwitz’s theorem](ch23-cards.md#g-hurwitz), [the octonions](ch23-cards.md#g-octonions)

### a Moufang identity, evaluated — `(e₄e₁)(e₂e₄) − e₄((e₁e₂)e₄)` {#g-moufang}

*Integer or rational*

0: the octonions obey Ruth Moufang’s identities, of which (zx)(yz) = z((xy)z) is one, and this instance vanishes even though e₁, e₂, e₄ do not associate.

Ruth Moufang introduced in 1935 the structures now called Moufang loops; the nonzero octonions, though not a group, satisfy identities such as (zx)(yz) = z((xy)z), z(x(zy)) = ((zx)z)y, and ((xz)y)z = x(z(yz)), and the unit octonions are the standard example. With x = e₁, y = e₂, z = e₄, whose ordinary associator is 2e₇, the Moufang difference is exactly 0 under the Cayley–Dickson table (verified by computation). Exactly 0, an integer, riding beside the alternative law.

Cites: [23.15](ch23-refs.md#r-23.15), [23.5](ch23-refs.md#r-23.5)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [the octonions](ch23-cards.md#g-octonions)

### the norm of the all-ones octonion, squared — `‖1 + e₁ + … + e₇‖^{2}` {#g-onorm}

*Integer or rational*

8: the sum of eight squares, and the octonion norm is multiplicative, which is the eight-square identity of Degen, Graves, and Cayley.

The octonion norm ‖x‖² = x x̄ = x₀² + … + x₇² is multiplicative, ‖xy‖ = ‖x‖‖y‖, and that statement is Degen’s eight-square identity, found by Carl Ferdinand Degen around 1818 and rediscovered by Graves in his letter to Hamilton of 26 December 1843 and by Cayley in 1845. Hurwitz proved in 1898 that such an identity exists only for 1, 2, 4, and 8 squares. For the all-ones octonion the norm squared is 8 (verified by computation). Value 8, an integer, a factor of the denominator.

Cites: [23.2](ch23-refs.md#r-23.2), [23.6](ch23-refs.md#r-23.6), [23.7](ch23-refs.md#r-23.7)

### the Fano plane — `PG(2, 2)` {#g-fano}

*Non-real object*

The smallest projective plane, seven points and seven lines with three points on each line: the multiplication table of the seven imaginary octonion units.

Gino Fano described the seven-point plane in 1892. Label its points e₁ … e₇ and orient each of its seven lines; then each line is a copy of the quaternion units i, j, k, and the whole octonion multiplication table is read off the picture. The book uses the Cayley–Dickson labeling, in which the lines are {1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6} (read off the computed products); Baez’s 2002 survey uses a different labeling, with the same geometry. Its symmetry group has order 168, which is also the number of signed zero-divisor pairs Chapter ᚗ(V) counts. A non-real object and a tool of the chapter.

Cites: [23.10](ch23-refs.md#r-23.10), [23.19](ch23-refs.md#r-23.19), [23.4](ch23-refs.md#r-23.4), [23.9](ch23-refs.md#r-23.9), [24.8](ch24-refs.md#r-24.8)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule)

### units of the integral octonions — `240` {#g-cayleyint}

*Integer or rational*

The integral octonions (Coxeter’s integral Cayley numbers) have exactly 240 elements of norm 1, and they sit at the 240 roots of E₈.

Dickson (1923) and Kirmse (1924) studied octonions with integer-like coordinates, Kirmse with an error that Coxeter corrected in 1946: there are seven maximal orders, not eight, all isometric to the E₈ lattice scaled by 1/√2, and each has exactly 240 elements of norm 1, forming a Moufang loop of order 240. Those 240 points are the root system of E₈, whose 248-dimensional Lie algebra is in Chapter ᚌ(G)’s denominator, and those same 240 vectors realize the kissing number in eight dimensions, which Levenshtein, and independently Odlyzko and Sloane, fixed at 240 in 1979. Viazovska proved the E₈ sphere packing itself optimal in 2016. A key entry, not in the factor. Value 240, an integer.

Cites: [23.11](ch23-refs.md#r-23.11), [23.12](ch23-refs.md#r-23.12)

See also: [the dimension of E₈](ch14-cards.md#g-e8), [a Lie algebra](ch14-cards.md#g-liealg), [the octonions](ch23-cards.md#g-octonions), [square root of two](ch04-cards.md#g-sqrt2)

### Hurwitz’s theorem — `1, 2, 4, 8` {#g-hurwitz}

*Integer or rational*

The only dimensions, 1, 2, 4, and 8, in which a normed division algebra over the reals exists: ℝ, ℂ, ℍ, 𝕆, and no more.

Adolf Hurwitz proved in 1898 that a sum-of-squares identity (x₁² + … + xₙ²)(y₁² + … + yₙ²) = z₁² + … + zₙ² with bilinear z exists only for n = 1, 2, 4, 8, and hence that the normed division algebras over ℝ are exactly ℝ, ℂ, ℍ, and 𝕆; the general classification of composition algebras appeared posthumously in 1923. Frobenius had shown in 1877 that the associative ones are ℝ, ℂ, ℍ. The Cayley–Dickson ladder keeps climbing past 8, but from the sedenions on the norm stops multiplying and zero divisors appear, which is Chapter ᚗ(V). A key entry, four integers.

Cites: [23.14](ch23-refs.md#r-23.14), [23.18](ch23-refs.md#r-23.18), [23.7](ch23-refs.md#r-23.7)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [the real numbers](ch01-cards.md#g-reals), [the sedenions](ch24-cards.md#g-sedenions), [sedenion zero divisors](ch24-cards.md#g-zd)

### the seven-dimensional cross product — `e₁ × e₂ = e₃` {#g-cross7}

*Non-real object*

The imaginary octonions carry a bilinear cross product, orthogonal to both factors and of length ‖x‖‖y‖ sin θ, x × y = Im(xy) for pure imaginary x, y; such a product exists only in three and seven dimensions.

A bilinear product on ℝⁿ that is orthogonal to its two factors and has length ‖x‖‖y‖ sin θ exists only for n = 3 and n = 7 (Brown and Gray 1967, Massey 1983; the restriction is related to Hurwitz’s theorem); the three-dimensional one comes from the quaternions and the seven-dimensional one from the octonions, as x × y = Im(xy) = ½(xy − yx) for imaginary x, y. Under the Cayley–Dickson table e₁ × e₂ = e₃ and, as always, x × x = 0; unlike its three-dimensional cousin it fails the Jacobi identity and is preserved only by G₂, not by all of SO(7). A non-real object and a tool of the chapter.

Cites: [23.13](ch23-refs.md#r-23.13)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [the dimension of G₂](ch14-cards.md#g-g2), [Hurwitz’s theorem](ch23-cards.md#g-hurwitz), [the Jacobi identity](ch14-cards.md#g-jacobi), [the octonions](ch23-cards.md#g-octonions), [the quaternions](ch22-cards.md#g-quaternions)

### the octonions — `𝕆 = {x₀ + x₁e₁ + ··· + x₇e₇ : xᵢ ∈ ℝ}` {#g-octonions}

*Non-real object*

The eight-dimensional real normed division algebra 𝕆: alternative but not associative, the last Cayley–Dickson algebra with a multiplicative norm and no zero divisors.

An octonion has eight real coordinates. The third Cayley–Dickson doubling gives the ladder ℝ → ℂ → ℍ → 𝕆: every nonzero octonion is invertible and the norm is multiplicative, but multiplication is not associative. The octonions remain alternative, so every subalgebra generated by two elements is associative. Hurwitz’s theorem makes 𝕆 the last real normed division algebra; the next doubling, the sedenions, has zero divisors and loses norm multiplicativity. A non-real object and the ambient number system of this chapter.

Cites: [23.1](ch23-refs.md#r-23.1), [23.14](ch23-refs.md#r-23.14), [23.3](ch23-refs.md#r-23.3), [23.7](ch23-refs.md#r-23.7)

See also: [the Cayley–Dickson doubling rule](ch24-cards.md#g-cdrule), [Hurwitz’s theorem](ch23-cards.md#g-hurwitz), [the sedenions](ch24-cards.md#g-sedenions), [sedenion zero divisors](ch24-cards.md#g-zd)