# References — Chapter ᚖ(W): The third landing *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-23, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch23.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.* ### 23.1 {#r-23.1} The octonions are an eight-dimensional real algebra and are nonassociative; their multiplication can be encoded by the seven-point Fano plane. *Status: confirmed* - [John C. Baez, The Octonions, Bulletin of the AMS 39 (2002)](https://math.ucr.edu/home/baez/octonions/) — Baez identifies the octonions as “the largest of the four normed division algebras,” “somewhat neglected due to their nonassociativity”; the Fano-plane multiplication mnemonic is not on this page but in §2.1, “The Fano Plane,” linked from its table of contents under “Constructing the Octonions.” - [John C. Baez, The Octonions, arXiv:math/0105155, introduction](https://arxiv.org/html/math/0105155v1) — the chapter epigraph is verbatim from the introduction, ‘The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.’ The book quotes the second sentence and trims its opening ‘But’. The survey appeared as Bulletin of the American Mathematical Society 39 (2002), 145–205. ### 23.2 {#r-23.2} Graves discovered the eight-dimensional algebra in December 1843 and described it to Hamilton in a letter of 26 December 1843; Cayley found it independently and got it into print first, in the Philosophical Magazine of March 1845; Hamilton vouched for Graves’s priority in a note of 14 June 1847. The eight-square identity itself had been found earlier still, by C. F. Degen in 1818. The doubling is the Cayley–Dickson construction. *Status: confirmed* - [John C. Baez, The Octonions, §2 Constructing the Octonions](https://math.ucr.edu/home/baez/octonions/node3.html) — gives the multiplication table and the index-cycling and doubling identities. The Graves/Cayley history is on node1, cited below in this same reference. - [Wikipedia, Octonion](https://en.wikipedia.org/wiki/Octonion) — gives the Cayley–Dickson construction and the independent Graves/Cayley history. - [John C. Baez, The Octonions, history](https://math.ucr.edu/home/baez/octonions/node1.html) — Graves wrote to Hamilton on 26 December 1843 describing the octaves (the page gives the letter’s date, not a day of discovery), sending three further letters in January 1844; Hamilton offered to publicize the work and kept postponing it; ‘In March of 1845, he [Cayley] published a paper in the Philosophical Magazine entitled “On Jacobi’s Elliptic Functions, in Reply to the Rev. B. Bronwin; and on Quaternions”’, with the octonions tacked on at the end; Graves added a priority postscript to a paper in the next issue of the same journal; and ‘On June 14th, 1847, Hamilton contributed a short note to the Transactions of the Royal Irish Academy, vouching for Graves’ priority’, by which time the name Cayley numbers had stuck. Re-read raw on 17 September 2026 for the Degen sentence, which the #onorm card and the p11 brace label both assert and which no entry carried: ‘Still worse, Graves later found that his eight squares theorem had already been discovered by C. F. Degen in 1818’. Editorial note: The chronology is no longer a judgement call: Baez’s history gives every step of it, and the page now says what the sources say. Graves found them first, in December 1843; Cayley found them independently and printed them first, in March 1845; Hamilton vouched for Graves’s priority on 14 June 1847, too late for the name. Cayley’s March 1845 octonion paper is a different paper from the February 1845 Philosophical Magazine paper cited at 22.8 for quaternion rotation, and the two must not be collapsed into one 1845 Cayley item. ### 23.3 {#r-23.3} The octonions are alternative: an associator vanishes when two arguments coincide, and Artin’s theorem says any two octonions generate an associative subalgebra. *Status: confirmed* - [John C. Baez, The Octonions, §1.1 Preliminaries (alternative laws and Artin’s theorem)](https://math.ucr.edu/home/baez/octonions/node2.html) — states the octonions are alternative and records Artin’s theorem that any two elements generate an associative subalgebra. - [Alternative algebra, Wikipedia](https://en.wikipedia.org/wiki/Alternative_algebra) — gives the left alternative law x(xy) = (xx)y, the right alternative law (yx)x = y(xx) and the flexible law (xy)x = x(yx), and states ‘Artin’s theorem states that in an alternative algebra the subalgebra generated by any two elements is associative’, crediting it to Artin and citing Schafer, with no publication year. Editorial note: The card used to add that Artin’s theorem was ‘published by Zorn in 1930 and credited by him to Artin’. That is true of the Artin–Zorn theorem — a finite alternative division ring is a field — and not of the two-generator theorem the card is about; Wikipedia and the nLab both credit the latter to Artin through Schafer and give it no year. The attribution to Artin stands; the year and the Zorn publication credit are gone. ### 23.4 {#r-23.4} Under the chapter’s Cayley–Dickson convention, (e₁e₂)e₄ − e₁(e₂e₄) = 2e₇, while the Fano-plane labeling in Baez’s survey assigns a different nonzero triple. *Status: confirmed* - **Own computation using the chapter’s Cayley–Dickson product** (the author's own work) — Computed from (a, b)(c, d) = (ac − d̄b, da + bc̄) and the quaternion basis: (e₁e₂)e₄ = e₇ and e₁(e₂e₄) = −e₇, hence the difference is 2e₇. The same product gives the seven lines {1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6}; under Baez’s table (e_i e_{i+1} = e_{i+3}, so e₁e₂ = e₄), e₁, e₂, e₄ lie on a line and (e₁e₂)e₃ − e₁(e₂e₃) = −2e₆ (recomputed 23 September 2026). - [John C. Baez, The Octonions, §2.1 The Fano plane](https://math.ucr.edu/home/baez/octonions/node4.html) — gives the Fano-plane labeling of the octonion multiplication table. The labeling is a convention, so relabeling changes which basis triple displays a chosen associator — that observation is this book’s, not the survey’s. - [John C. Baez, The Octonions, §2 Constructing the Octonions (multiplication table)](https://math.ucr.edu/home/baez/octonions/node3.html) — the index-cycling and doubling rules, “Together with a single nontrivial product like e₁e₂ = e₄, these facts are enough to recover the whole multiplication table.” ### 23.5 {#r-23.5} Moufang identities include (zx)(yz) = z((xy)z), and the unit octonions form the standard Moufang loop despite octonion multiplication being nonassociative. *Status: confirmed* - [John C. Baez, The Octonions, §3 Octonionic Projective Geometry](https://math.ucr.edu/home/baez/octonions/node8.html) — cited for Ruth Moufang’s 1933 non-Desarguesian projective plane: “in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions”; the same page says a projective plane comes from a skew field if and only if it satisfies “the axiom of Desargues.” The survey never states the Moufang identities and never uses the word loop; those are carried by the Wikipedia entry below. - [EUDML record for Moufang, Zur Struktur von Alternativkörpern (1935)](https://eudml.org/doc/159732) — bibliographic record confirms Moufang’s 1935 paper on the structure of alternative fields. - [Moufang loop, Wikipedia](https://en.wikipedia.org/wiki/Moufang_loop) — opens ‘Moufang loops were introduced by Ruth Moufang (1935)’, lists the identities z(x(zy)) = ((zx)z)y, x(z(yz)) = ((xz)y)z, (zx)(yz) = (z(xy))z and (zx)(yz) = z((xy)z), and states ‘The nonzero octonions form a nonassociative Moufang loop under octonion multiplication’; and, of alternative algebras, “Multiplication in an alternative ring or alternative algebra R obey the Moufang identities.” ### 23.6 {#r-23.6} The octonion norm is multiplicative, giving the eight-square composition identity; the all-ones octonion has squared norm 8. Graves showed in his letter of 26 December 1843 that the octaves form a normed division algebra. The eight-square identity was first found by Carl Ferdinand Degen around 1818 and rediscovered by Graves (1843) and Cayley (1845). *Status: confirmed* - [John C. Baez, The Octonions, normed division algebras](https://math.ucr.edu/home/baez/octonions/node2.html) — states the normed-division-algebra norm is multiplicative and identifies the octonions as the eight-dimensional case. - **Own computation** (the author's own work) — For 1 + e₁ + ··· + e₇, the coordinate norm square is 1² + seven·1² = 8. - [John C. Baez, The Octonions, §1.1 Preliminaries (history)](https://math.ucr.edu/home/baez/octonions/node1.html) — “On December 26th, he wrote to Hamilton describing a new 8-dimensional algebra, which he called the ‘octaves’. He showed that they were a normed division algebra.” - [Degen’s eight-square identity, Wikipedia](https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity) — read as raw wikitext: “First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845).” It opens: “Degen’s eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares.” ### 23.7 {#r-23.7} Hurwitz’s theorem restricts real normed division algebras to dimensions 1, 2, 4 and 8, namely the reals, complexes, quaternions and octonions. Two events: the 1, 2, 4, 8 sums-of-squares theorem is Hurwitz’s Göttingen paper of 1898; the classification of the normed composition algebras is his posthumous paper in Mathematische Annalen 88 (1923). *Status: confirmed* - [Geometry of the Octonions, The Hurwitz Theorem](https://octonions.geometryof.org/GO/book/go/hurwitz.html) — states that the only real composition/normed division algebras with positive-definite norm are R, C, H and O, of dimensions 1, 2, 4 and 8. - [John C. Baez, The Octonions, normed division algebras](https://math.ucr.edu/home/baez/octonions/node2.html) — dates the underlying Hurwitz result to 1898 and states the four-algebra classification. - [Hurwitz’s theorem (composition algebras), Wikipedia](https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)) — ‘Hurwitz’s theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898’, citing A. Hurwitz, ‘Über die Composition der quadratischen Formen von beliebig vielen Variabeln’, Goett. Nachr. (1898), 309–316; the general composition-algebra theorem is the posthumous paper, Math. Ann. 88 (1923), 1–25. Editorial note: Hurwitz has two papers here and, in this book, two theorems. The 1, 2, 4, 8 result the chapter uses is the Göttingen paper of 1898, published in 1898. The card previously read ‘proved in 1898 (published 1923)’, which is wrong: 1923 is the posthumous Mathematische Annalen paper, a different and more general piece of work. The card now says 1898, which is what the chapter prose says and what Chapter ᚔ(I) says at 21.14. Hurwitz’s quaternion integers of 1896/1919, cited in Chapter ᚕ(J) at 22.7, are a third thing again. ### 23.8 {#r-23.8} The compact exceptional group G₂ is the automorphism group of the octonions, has dimension 14, and is the smallest of the exceptional Lie groups. *Status: confirmed* - [Geometry of the Octonions, G2](https://sites.science.oregonstate.edu/coursewikis/GO/dokue6fd.html?id=book%3Acontent%3Ag2&rev=1411236840) — identifies Aut(O) with G₂ and counts 14 independent automorphisms. - [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’, and derives dim G₂ = 14 from Spin(7)/G₂ = S⁷. ### 23.9 {#r-23.9} The Fano plane is the projective plane over the two-element field, with seven points and seven lines, three points on each line; oriented lines encode octonion products. *Status: confirmed* - [John C. Baez, The Octonions, §2.1 (author’s own HTML)](https://math.ucr.edu/home/baez/octonions/node4.html) — the Fano plane has 7 points and 7 lines, three points to a line, is the projective plane over Z₂, and gives the rule e_i e_j = e_k for cyclically ordered triples (the paper: Bull. AMS 39 (2002) 145–205, doi:10.1090/S0273-0979-01-00934-X; also arXiv:math/0105155). ### 23.10 {#r-23.10} The Fano plane was described by Gino Fano in 1892, and its automorphism group has order 168. Fano’s 1892 paper on the foundations of projective geometry constructs the model of a projective plane now named after him. *Status: confirmed* - [Wikipedia, Fano plane](https://en.wikipedia.org/wiki/Fano_plane) — gives the seven-point/seven-line plane, its order-168 automorphism group, and the historical attribution to Gino Fano; the Fano plane “is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.” - [Finite geometry, Wikipedia](https://en.wikipedia.org/wiki/Finite_geometry) — ‘In 1892, Gino Fano was the first to consider such a finite geometry’, citing Fano, ‘Sui postulati fondamentali della geometria proiettiva’, Giornale di Matematiche 30 (1892), 106–132. - [A. Collino, A. Conte, A. Verra, On the life and scientific work of Gino Fano, arXiv:1311.7177](https://arxiv.org/abs/1311.7177) — read from the PDF: Fano’s ideas “are neatly expressed in his paper of 1892 on foundations of projective geometry, where a famous model of projective plane, named today the Fano plane, is in particular constructed.” Editorial note: The Fano plane article carries the seven points and seven lines and the collineation group of order 168 = 2³·3·7, but it does not give a year for Fano. The year 1892, which the card asserts, comes from the finite-geometry article and the Fano paper it cites. ### 23.11 {#r-23.11} The Coxeter integral octonions have 240 norm-one elements, corresponding to the 240 roots of E₈ and forming a Moufang loop; those 240 roots also realize the eight-dimensional kissing number. E₈ is a simple Lie algebra and Lie group of dimension 248. *Status: confirmed* - [John C. Baez, Integer Octonions](https://math.ucr.edu/home/baez/octonions/integers/) — links the 240 smallest integral octonions with the E₈ root polytope. - [Wikipedia, Octonion](https://en.wikipedia.org/wiki/Octonion) — gives Kirmse (1924) and Coxeter (1946), records that Kirmse wrongly counted eight maximal orders where there are seven because his integers are not closed under multiplication, and states that each of the seven has 240 elements of minimum nonzero norm 1 forming a Moufang loop of order 240: “The Kirmse integers and the seven maximal orders are all isometric to the E8 lattice rescaled by a factor of 1/√2.” - [Kissing number, Wikipedia](https://en.wikipedia.org/wiki/Kissing_number) — the kissing number in eight dimensions is 240, realized by the E₈ lattice, and was settled by Levenshtein (1979) and independently by Odlyzko and Sloane (1979). - [Holger P. Petersson, Integral octonions (2019)](https://www.fernuni-hagen.de/mi/fakultaet/emeriti/docs/petersson/ass.-rem.-int.-oct.pdf) — Dickson’s ‘A new simple theory of hypercomplex integers’ of 1923 already constructs an algebra of integral octonions isomorphic to Coxeter’s of 1946, so the Coxeter order is more than twenty years older than is usually assumed. - [E8 (mathematics), Wikipedia](https://en.wikipedia.org/wiki/E8_(mathematics)) — read as raw wikitext: the smallest nontrivial representation of E₈ is “the adjoint representation (of dimension 248) acting on the Lie algebra E₈ itself.” Editorial note: The card’s dated names all sit here now: Dickson 1923 from Petersson, Kirmse 1924 and Coxeter 1946 and the seven-not-eight maximal orders from the Octonion article, and the kissing number from its own article. The kissing number used to ride inside a parenthesis whose only marker was 23.12, which is Viazovska on packing density — a different theorem, proved by different people in a different decade. ### 23.12 {#r-23.12} Viazovska proved in 2016 that the E₈ lattice gives the optimal sphere packing in eight dimensions. *Status: confirmed* - [Maryna Viazovska, The sphere packing problem in dimension 8](https://arxiv.org/abs/1603.04246) — the 2016 paper’s abstract states that no packing of unit balls in R⁸ has density greater than the E₈-lattice packing. ### 23.13 {#r-23.13} A norm-compatible vector cross product exists only in dimensions 3 and 7; the seven-dimensional one is obtained from the imaginary octonions and is preserved by G₂ rather than all of SO(7). *Status: confirmed* - [Brown and Gray, Vector Cross Products, Commentarii mathematici Helvetici 42 (1967)](https://eudml.org/doc/139340) — bibliographic record for the primary classification paper on vector cross products. - [John C. Baez, The Octonions, G2 and the seven-dimensional cross product](https://math.ucr.edu/home/baez/octonions/node14.html) — relates the imaginary octonions to the seven-dimensional cross product and G₂ symmetry. - [Seven-dimensional cross product, Wikipedia](https://en.wikipedia.org/wiki/Seven-dimensional_cross_product) — ‘Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions’; cites Robert B. Brown and Alfred Gray, ‘Vector cross products’, Commentarii Mathematici Helvetici 42 (1967), 222–236, and W. S. Massey, ‘Cross products of vectors in higher dimensional Euclidean spaces’, The American Mathematical Monthly 90 (1983), 697–701; states that the seven-dimensional product does not satisfy the Jacobi identity and is invariant under G₂ rather than SO(7); “The restriction to 0, 1, 3 and 7 dimensions is related to Hurwitz’s theorem, that normed division algebras are only possible in 1, 2, 4 and 8 dimensions.” ### 23.14 {#r-23.14} The Cayley–Dickson process continues past the octonions to sedenions, where norm multiplicativity fails and zero divisors occur. The sedenions are not alternative and hence not a composition algebra: the norm is no longer multiplicative. *Status: confirmed* - [Wikipedia, Sedenion](https://en.wikipedia.org/wiki/Sedenion) — states that sedenions have zero divisors and are no longer a division algebra. - [John C. Baez, The Octonions, §2.2 The Cayley–Dickson construction](https://math.ucr.edu/home/baez/octonions/node5.html) — the construction “provides a nice way to obtain the sequence” of the four normed division algebras (printed as an image); the next algebra “is called the sedenions,” and “the sedenions, and thus all the rest, have zero divisors.” - [Cayley–Dickson construction, Wikipedia](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) — read as raw wikitext: the sedenions retain power associativity “but loses the property of being an alternative algebra and hence cannot be a composition algebra.” ### 23.15 {#r-23.15} The displayed Moufang and associator values under the chosen table are 0 and 2e₇ respectively, and the denominator evaluates to 224e₇ after the associator is parenthesized as a factor. *Status: confirmed* - **Own computation under the chapter’s Cayley–Dickson convention** (the author's own work) — Computed the two numerator differences and denominator: 2e₇ · 8 · 14 = 224e₇; the repeated-entry alternative-law term and the Moufang difference both evaluate to 0. ### 23.16 {#r-23.16} Moufang’s identities were developed in connection with projective-plane work in the 1930s. *Status: judgement* - [John C. Baez, The Octonions, projective planes](https://math.ucr.edu/home/baez/octonions/node8.html) — dates Moufang’s remarkable non-Desarguesian projective-plane example to 1933. - [EUDML record for Moufang (1935)](https://eudml.org/doc/159732) — records the 1935 paper on alternative fields associated with the Moufang identities. - [Moufang loop, Wikipedia](https://en.wikipedia.org/wiki/Moufang_loop) — opens ‘Moufang loops were introduced by Ruth Moufang (1935)’, lists the identities z(x(zy)) = ((zx)z)y, x(z(yz)) = ((xz)y)z, (zx)(yz) = (z(xy))z and (zx)(yz) = z((xy)z), and states ‘The nonzero octonions form a nonassociative Moufang loop under octonion multiplication’. Editorial note: Two papers, one year on the page. Baez dates the non-Desarguesian projective plane to 1933; the identities themselves are the 1935 paper, and the Moufang loop article dates their introduction to 1935 as well. So the page’s ‘found the identities in 1935 while studying projective planes’ has both halves behind it, and what stays a judgement is only the compression of a two-paper story into one year. ### 23.17 {#r-23.17} The octonions occur in connections with mathematical physics, including string-theoretic work. *Status: judgement* - [John C. Baez, The Octonions](https://math.ucr.edu/home/baez/octonions/) — abstract lists connections to spinors, Bott periodicity, exceptional Lie groups, and supersymmetry. Editorial note: The mathematical connections are established, but the prose’s phrase ‘in a way nobody has fully explained’ is interpretive. We cite the connections and leave the explanatory-status wording as authorial voice. ### 23.18 {#r-23.18} Frobenius classified the finite-dimensional associative division algebras over the reals as ℝ, ℂ and ℍ; the proof is dated 1877 and the paper appeared in Crelle’s Journal in 1878. *Status: confirmed* - [Frobenius theorem (real division algebras), Wikipedia](https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)) — opens ‘the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877’, characterizes the finite-dimensional associative division algebras over the reals as the reals, the complex numbers and the quaternions, and cites F. G. Frobenius (1878), ‘Über lineare Substitutionen und bilineare Formen’, Journal für die reine und angewandte Mathematik 84, 1–63. Editorial note: The #hurwitz card’s Frobenius sentence stood on 23.7, which is Hurwitz’s theorem and does not mention him. They are different classifications and easy to swap: Frobenius assumes associativity and no norm, Hurwitz assumes a multiplicative norm and no associativity, which is why the octonions fall out of one and not the other. Hence its own entry rather than a stretch of 23.7. The source dates the proof to 1877 and the printed paper to 1878; the card says 1877, and both are recorded here so a later pass does not read one as a correction of the other. ### 23.19 {#r-23.19} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Graves’s octonions (‘octaves’), a normed division algebra, 1843; Baez’s survey ‘The Octonions’ is dated 2002. *Status: confirmed* - [Octonion, Wikipedia](https://en.wikipedia.org/w/index.php?title=Octonion) — read raw: “The octonions were discovered in December 1843 by John T. Graves, inspired by his friend William Rowan Hamilton’s discovery of quaternions..” - [The Octonions, John C. Baez (author’s page)](https://math.ucr.edu/home/baez/octonions/) — read raw: “Published in Bull. Amer. Math. Soc. 39 (2002), 145-205..” 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.