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# References — Chapter ᚗ(V): The top landing

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-24, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch24.md)

*Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.*

### 24.1 {#r-24.1}

The Cayley–Dickson construction doubles dimension at each step: from octonions it produces the 16-dimensional sedenions and can continue to dimensions 32, 64, and every power of two.

*Status: confirmed*

- [Wikipedia, Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) — describes repeated doubling, dimension 2^n, and the sequence through sedenions and beyond.

### 24.2 {#r-24.2}

The real, complex, quaternion and octonion composition/division algebras have no zero divisors; sedenions do, so nonzero elements can multiply to zero and cancellation and division fail. The counts of two-unit-sum zero divisors (42 supports, 84 splittings, 168 signed pairs, 336 ordered) are the book’s own enumeration, set out at 24.8.

*Status: confirmed*

- [Baez, The Octonions](https://math.ucr.edu/home/baez/octonions/node2.html) — states that the only real normed division algebras have dimensions 1, 2, 4 and 8 and explains the multiplicative norm/division property.
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — identifies sedenions as 16-dimensional and states that they have zero divisors, unlike the lower Cayley–Dickson algebras.

### 24.3 {#r-24.3}

Under the chapter’s Cayley–Dickson sign convention, (e₃ + e₁₀)(e₆ − e₁₅) is zero, while both factors are nonzero; the product has zero in all sixteen coordinates. Its four unit products are e₃e₆ = e₅, e₃e₁₅ = e₁₂, e₁₀e₆ = e₁₂ and e₁₀e₁₅ = e₅, and for a = e₃ + e₁₀, b = e₆ − e₁₅, a² = −2, so (aa)b = −2b while a(ab) = 0.

*Status: confirmed*

- **Own computation** (the author's own work) — Implemented the displayed Cayley–Dickson pair-product rule and the chapter’s basis-index convention in an exact integer-coordinate calculation: the two 16-coordinate factors are nonzero and their product is the all-zero vector. The same calculation gives e₃e₆ = e₅, e₃e₁₅ = e₁₂, e₁₀e₆ = e₁₂ and e₁₀e₁₅ = e₅, so ab = e₅ − e₁₂ + e₁₂ − e₅ = 0; a² = −2, so a has the two-sided inverse −a/2; and (aa)b = −2b while a(ab) = 0, so the regrouping a⁻¹(ab) = (a⁻¹a)b, an instance of the alternative law, fails for this pair (re-run 29 September 2026).
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — gives explicit nontrivial zero-divisor examples of the two-unit-sum form in the standard sedenion basis.

### 24.4 {#r-24.4}

The displayed sedenion zero-divisor factors each have norm √2, so their norm product is 2, while the norm of their zero product is 0.

*Status: confirmed*

- **Own computation** (the author's own work) — Each factor has exactly two coefficients ±1 and fourteen zero coefficients, so the Euclidean sedenion norm is sqrt(1²+1²)=√2; their product is 0 by citation 24.3, whose norm is 0, and √2·√2=2.
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — identifies the sedenions as the 16-dimensional Cayley–Dickson algebra and records that they have zero divisors and are not a normed division algebra. It does not define the norm or discuss multiplicativity; the norm arithmetic above is this book’s own.

### 24.5 {#r-24.5}

Hurwitz’s theorem restricts real normed division/composition algebras with multiplicative positive norm to dimensions 1, 2, 4 and 8; the result is associated with Hurwitz’s 1898 work.

*Status: confirmed*

- [Baez, The Octonions](https://math.ucr.edu/home/baez/octonions/node2.html) — states the four possible dimensions and identifies the first theorem with Hurwitz’s 1898 paper.
- [EUDML record, Über die Composition der quadratischen Formen von beliebig vielen Variablen](https://eudml.org/doc/58420) — bibliographic record for Hurwitz’s 1898 composition paper.

### 24.6 {#r-24.6}

Sedenions are noncommutative, nonassociative, nonalternative, but power-associative; all Cayley–Dickson algebras are flexible and power-associative. Schafer’s 1954 result is the flexibility; the power associativity of every Cayley–Dickson algebra is stated by the Cayley–Dickson construction page without attribution.

*Status: confirmed*

- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — “the sedenions form a 16-dimensional noncommutative and nonassociative algebra”; “Unlike the octonions, the sedenions are not an alternative algebra”; its infobox lists power associativity among the main properties.
- [Wikipedia, Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) — states that the construction produces power-associative algebras and cites Schafer for flexibility.

### 24.7 {#r-24.7}

The computation (aa)a − a(aa) = 0 for a = e₃ + e₁₀ is a valid instance of power associativity; the flexible identity is (ab)a = a(ba).

*Status: confirmed*

- **Own computation** (the author's own work) — Using the same exact sedenion multiplication table as citation 24.3, both bracketings of a·a·a for a=e₃+e₁₀ give the same 16-coordinate vector, so their difference is zero.
- [Wikipedia, Flexible algebra](https://en.wikipedia.org/wiki/Flexible_algebra) — defines the flexible identity and attributes the Cayley–Dickson result to Schafer (1954).

### 24.8 {#r-24.8}

The simple two-unit-sum zero-divisor enumeration has 42 four-unit supports, 84 unordered pair splittings, 168 signed unordered factor pairs, and 336 when factor order is counted; no support contains the real unit, and each support splits into a vanishing product in exactly two of its three possible ways.

*Status: confirmed*

- **Own enumeration** (the author's own work) — Enumerated all pairs (e_i ± e_j)(e_k ± e_l) with distinct imaginary indices 1≤i<j, k<l≤15 under the chapter’s Cayley–Dickson table. Deduplicating by unordered factors and ignoring signs gives 84 splittings; deduplicating supports gives 42 four-element sets; each splitting has two vanishing sign patterns, giving 168 signed unordered pairs and 336 ordered pairs. The chapter’s support {3, 6, 10, 15} vanishes as (e₃ + e₁₀)(e₆ − e₁₅) and as (e₃ + e₁₅)(e₆ + e₁₀), and not in its third splitting (re-run 29 September 2026).
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — its table lists 84 zero-divisor pairs of the two-unit-sum form among the fifteen imaginary units.
- [Robert de Marrais, The 42 Assessors and the Box-Kites they fly: Diagonal Axis-Pair Systems of Zero-Divisors in the Sedenions’ 16 Dimensions, arXiv:math/0011260](https://arxiv.org/abs/math/0011260) — re-checked at arXiv:math/0011260 (submitted 30 November 2000): the abstract reads ‘The 168 are arranged in point-set quartets along the 42 Assessors’, which reaches the same two numbers by another route. The identification is not an identity and the entry no longer claims one: de Marrais’s 42 Assessors are pairs of diagonals in planes each spanned by two units, carrying 168 unit zero divisors four to a plane, whereas the enumeration above counts 42 four-unit supports and 168 signed unordered factor pairs. The counts agree; the objects counted differ.

### 24.9 {#r-24.9}

The norm-one sedenion zero-divisor pairs form a space associated with the exceptional Lie group G₂, the automorphism group of the octonions.

*Status: confirmed*

- [Holger P. Petersson, The Non-Orthogonal Cayley–Dickson Construction and the Octonionic Structure of the E₈-Lattice](https://www.fernuni-hagen.de/mi/fakultaet/emeriti/docs/petersson/north-cd-e8.pdf) — (§9.9): “the zero divisor pairs ((x, y) ∈ S² satisfying ‖x‖ = ‖y‖ = 1 and xy = 0) form a principal homogeneous space for the compact Lie group G₂,” citing Moreno 1998. Petersson carries the principal-homogeneous-space result only; he does not establish the separate identification of G₂ with Aut(𝕆).
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — carries Moreno’s homeomorphism result. It says nothing about automorphisms or octonion symmetry; that half of the claim rests on the Baez source below.
- [John C. Baez, The Octonions, exceptional Lie groups](https://math.ucr.edu/home/baez/octonions/node14.html) — ‘the smallest of the exceptional Lie groups, G₂, is the automorphism group of the octonions’. Carries the automorphism-group half of this reference’s claim; the same page and quotation already stand at r-23.8.

### 24.10 {#r-24.10}

The Cayley–Dickson doubling rule used by the chapter builds the sequence from ℝ to ℂ, ℍ, 𝕆, 𝕊 and then higher 2^n-dimensional algebras; the general construction is due to Leonard Dickson in 1919, who showed the octonions as a two-dimensional algebra over the quaternions, and it is named for Arthur Cayley and Dickson.

*Status: confirmed*

- [Wikipedia, Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) — gives the pair construction, conjugation, dimension doubling, and the standard real-to-complex-to-quaternion-to-octonion sequence, and reads ‘The Cayley–Dickson construction is due to Leonard Dickson in 1919 showing how the octonions can be constructed as a two-dimensional algebra over quaternions’ and ‘It is named after Arthur Cayley and Leonard Eugene Dickson’. It says nothing about Hamilton building the quaternions from pairs of complex numbers.

### 24.11 {#r-24.11}

Doubling the sedenions gives the 32-dimensional trigintaduonions, and the construction continues to dimensions 64, 128 and beyond; higher Cayley–Dickson algebras retain distributivity, power associativity and flexibility but have zero divisors; the trigintaduonions have 1,260 of them, the same count the sedenion enumeration gives when it is run in 32 dimensions.

*Status: confirmed*

- [Wikipedia, Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) — confirms indefinite dimension doubling and the preservation of power associativity and flexibility.
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — states that higher Cayley–Dickson algebras have zero divisors and identifies the next 32-dimensional algebra as the trigintaduonions.
- [Wikipedia, Trigintaduonion](https://en.wikipedia.org/wiki/Trigintaduonion) — ‘Whereas the sedenions have 84 zero divisors, the trigintaduonions have 1,260 zero divisors derived from 147 distinct triples’, and states that they are power-associative, flexible and distributive but not alternative.
- **Own enumeration** (the author's own work) — The same exhaustive count run at dimension 32 gives 630 four-unit supports and 1,260 splittings, matching the reported 1,260; the 147 triples are a different decomposition of the same set.

### 24.12 {#r-24.12}

The 84 counts pair splittings, not factor pairs taken up to order and an overall sign: (e₁ + e₁₂)(e₂ + e₁₅) and (e₁ − e₁₂)(e₂ − e₁₅) both vanish and are not related by an overall sign, so the sign quotient does not give 84.

*Status: confirmed*

- **Own enumeration** (the author's own work) — The exhaustive count distinguishes support, splitting, sign and factor order. Of the four sign patterns on the splitting {1, 12} | {2, 15}, exactly (+, +) and (−, −) vanish, and those two factor pairs are not related by an overall sign, so quotienting by sign gives 168 and not 84.
- [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — supplies the 84 two-unit-sum zero-divisor table that the chapter’s count is refining.

Editorial note: The counts 84, 168 and 336 are all correct; the equivalence relation named beside them was not. The 84 is the number of ways a four-unit support splits into two factors: 42 supports, two splittings each. Each splitting vanishes for exactly two of its four sign patterns, which is where 168 signed unordered pairs and 336 ordered ones come from. That is the reading both formula tips, the chapter key, the prose and the cards #zd and #zd84 all state.

### 24.13 {#r-24.13}

The displayed zero products and the power-associator difference are computed in the sedenion algebra and return the rational/integer value 0; the denominator value 2·84 is 168 and is nonzero.

*Status: confirmed*

- **Own computation** (the author's own work) — Exact integer arithmetic gives zero for both numerator terms, √2·√2=2, 2·84=168, and 0/168=0. The arithmetic is internal to the displayed formula.

### 24.14 {#r-24.14}

The chapter epigraph is quoted verbatim from Baez’s 2002 octonions survey, section 2.2 on the Cayley–Dickson construction, where it lists what each doubling costs.

*Status: confirmed*

- [John C. Baez, The Octonions, section 2.2, The Cayley–Dickson Construction](https://math.ucr.edu/home/baez/octonions/node5.html) — ‘each time we apply the construction, our algebra gets a bit worse. First we lose the fact that every element is its own conjugate, then we lose commutativity, then we lose associativity, and finally we lose the division algebra property.’ The chapter quotes the second sentence in full and unaltered.
- [John C. Baez, The Octonions, Bulletin of the AMS 39 (2002)](https://math.ucr.edu/home/baez/octonions/) — the author’s own page for the survey, giving ‘Published in Bull. Amer. Math. Soc. 39 (2002), 145-205’, which is the attribution the epigraph footer carries.

### 24.15 {#r-24.15}

Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Cayley printed the octonions in 1845.

*Status: confirmed*

- [The Octonions, §1 Introduction, John C. Baez](https://math.ucr.edu/home/baez/octonions/node1.html) — read raw: “In March of 1845, he published a paper in the Philosophical Magazine entitled ‘On Jacobi’s Elliptic Functions, in Reply to the Rev. B. Bronwin; and on Quaternions’.”

Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.