References

[Baez] J. C. Baez, “The Octonions”, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145–205; errata 42 (2005) 213. doi:10.1090/S0273-0979-01-00934-X; arXiv:math/0105155 (v4 checked; open copy). Used: §1.1, Thm 1 (Hurwitz) — row, Albert; §3, the list of simple formally real Jordan algebras — Albert; §3.4, determinant on \(\mathfrak h_3(\Oc)\) — Albert; §4.2, \(\Ffour=\Aut(\mathfrak h_3(\Oc))\), \(\ffour=\Der\) — Prop.; §4.3, Table 5 (magic square) — history; §1, pp. 1–3 (Hamilton, Graves, Cayley; SU(2) and rotations; \(\slf(2,\K)\cong\so(\dim\K+1,1)\); “still no proof that the octonions are useful for understanding the real world”) — quaternions and octonions, physics; §2.1, pp. 6–7 (Fano plane), §2.2, Props. 1–5, p. 9 (Cayley–Dickson) — Fano plane, doubling; §4.4, p. 46 (27-dimensional representations; \(E_{6(-26)}\) with \(\Ffour\) maximal compact) — octonions over \(\C\), physics. The rule \(e_ie_{i+1}=e_{i+3}\) is not printed literally; it follows from the index-cycling identity and \(e_1e_2=e_4\) of §2, p. 5.

Convention note. Baez §3.4 writes \(\begin{pmatrix}\alpha&z^*&y^*\\ z&\beta&x\\ y&x^*&\gamma\end{pmatrix}\) and \(\det=\alpha\beta\gamma-(\alpha\|x\|^2+\beta\|y\|^2+\gamma\|z\|^2)+2\Re(xyz)\). With the multiplication rule \(e_ie_{i+1}=e_{i+3}\) of Baez §2, our numerical test of the trace identity \(\det a=\tfrac13\tr a^3-\tfrac12\tr a^2\tr a+\tfrac16(\tr a)^3\), printed in the same section, holds for that matrix only with \(2\Re(xyz^*)\); equivalently, with \(z\) (not \(z^*\)) in position \((1,2)\), as on these pages, the term is \(\Tr((xy)z)\). Script: assets/figures/check_albert.py. Whether the 2005 errata addresses this was not checked.

[CS] C. Chevalley, R. D. Schafer, “The exceptional simple Lie algebras \(F_4\) and \(E_6\)”, Proc. Nat. Acad. Sci. USA 36 (1950), no. 2, 137–141. doi:10.1073/pnas.36.2.137; PMC1063148. Used: \(\ffour=\Der(\Jalg)\), \(\esix=\Der(\Jalg)+\{R_Y:\tr Y=0\}\) — Prop.. Content checked via the zbMATH review (Zbl 0037.02003) only; full text not opened.

[Yokota] I. Yokota, “Exceptional Lie groups”, arXiv:0902.0431 (v1 checked; printed arXiv pages). Used: §2.13, Definition, p. 65 (\(\Ffour^\C=\{\alpha:\det(\alpha X)=\det X,\ \alpha E=E\}\)) — Albert, Prop.; §3.1, Definition, p. 68 (\(\Esix^\C\)); Thm 3.2.1 (\(\esix^\C=\{\delta+\tilde T\}\), dim 78); Thm 3.14.1, p. 104 (\(\Esix^\C\simeq\Esix\times\R^{78}\)). Also Prop. 2.4.6 (\(\Jo\) is \(\ffour\)-irreducible) and Prop. 3.3.2 (\(\Jalg\) is \(\esix\)-irreducible). A 2025 book edition (Springer LNM 2369) exists; its numbering was not checked.

[Krutelevich] S. Krutelevich, “Jordan algebras, exceptional groups and Bhargava composition”, J. Algebra 314 (2007), no. 2, 924–977. doi:10.1016/j.jalgebra.2007.02.060; arXiv:math/0411104 (v1 checked, titled “… higher composition laws”). Used: Remark 10(iv), p. 15 (\(\mathrm{NP}(\mathfrak h_3(\Oc))\) simply connected of type \(E_6\)) — Prop.; Prop. 12(i), p. 16 (transitivity on elements of a given nonzero norm) — Prop.. Numbering in the journal version not checked. The step “over \(\C\) the octonions are split” is standard and not checked in a source.

[SV] T. A. Springer, F. D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Math., Springer, Berlin, 2000, viii+208 pp. doi:10.1007/978-3-662-12622-6. Used: Ch. 1 (composition algebras, pp. 1–23) for the trace identities — Albert; Ch. 7 (exceptional groups, pp. 173–183): automorphism group of type \(F_4\), invariance group of \(\det\) of type \(E_6\) — Albert. Thm 1.8.1 (over any field, one split composition algebra in each dimension 2, 4, 8) — octonions over \(\C\), location taken from Nagy–Vojtěchovský, arXiv:math/0701707, not from the book. Section and lemma numbers unverified (book not opened; content from the zbMATH review Zbl 1087.17001).

[JNW] P. Jordan, J. von Neumann, E. Wigner, “On an algebraic generalization of the quantum mechanical formalism”, Ann. of Math. (2) 35 (1934), no. 1, 29–64. doi:10.2307/1968117. Used: the classification — Albert, history, physics. Content checked via the zbMATH review (Zbl 0008.42103).

[Albert] A. A. Albert, “On a certain algebra of quantum mechanics”, Ann. of Math. (2) 35 (1934), no. 1, 65–73. doi:10.2307/1968118. Used: history, physics. Content checked via the zbMATH review (Zbl 0008.42104).

[BD] O. Biquard, T. Delcroix, “Ricci flat Kähler metrics on rank two complex symmetric spaces”, J. Éc. polytech. Math. 6 (2019), 163–201. doi:10.5802/jep.91; arXiv:1807.07129; open PDF. Used: §3.3, Table 1, p. 173 (AI, A2, AII, EIV: \(R=A_2\), multiplicities 1, 2, 4, 8; group case written \(\mathrm{PGL}_3\times\mathrm{PGL}_3/\mathrm{PGL}_3\)) — row.

[Klein] S. Klein, “Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2”, Osaka J. Math. 47 (2010), no. 4, 1077–1157; arXiv:0809.1319 (v1 checked). Used: §4.1, pp. 38–39 of the arXiv version (restricted roots of \(E_6/F_4\): type \(A_2\), all multiplicities 8; compact Riemannian setting) — Prop., row.

[Helgason] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, 1978 (Pure Appl. Math. 80); reprinted AMS Grad. Stud. Math. 34, 2001. Ch. X, pp. 438–537 in both editions (doi:10.1090/gsm/034/10). Used: the Satake diagram and root data of EIV — Prop.. Table number (VI) and page (about 532) inferred from citations in other papers; the book was not opened.

[Satake] I. Satake, “On representations and compactifications of symmetric Riemannian spaces”, Ann. of Math. (2) 71 (1960), no. 1, 77–110. doi:10.2307/1969880. Used: history.

[Araki] S. Araki, “On root systems and an infinitesimal classification of irreducible symmetric spaces”, J. Math. Osaka City Univ. Ser. A 13 (1962), 1–34. Zbl 0123.03002 (no DOI). Used: history. Full text not opened.

[Cartan 1926, 1927] É. Cartan, “Sur une classe remarquable d'espaces de Riemann”, Bull. Soc. Math. France 54 (1926), 214–264, doi:10.24033/bsmf.1105; II, 55 (1927), 114–134, doi:10.24033/bsmf.1113 (volumes from zbMATH; Crossref lists them wrongly). Used: history.

[Vust] T. Vust, “Opération de groupes réductifs dans un type de cônes presque homogènes”, Bull. Soc. Math. France 102 (1974), 317–333. doi:10.24033/bsmf.1782; open PDF. Used: §1.3, Thm 1 and the remark after it, p. 322 (\(K^\circ P\) open; Borel subgroups with \(K^\circ\Bor\) open) — spherical.

[Pezzini] G. Pezzini, “Lectures on spherical and wonderful varieties”, Les cours du CIRM 1 (2010), no. 1, 33–53. doi:10.5802/ccirm.3; numdam. Used: Def. 2.1.1 (spherical) — spherical; Ex. 2.2.5, p. 37 (\(\SL_3/\SO_3\cdot Z(\SL_3)\)) — spherical. It has no general numbered statement that symmetric varieties are spherical; for that see [Vust].

[Freudenthal] H. Freudenthal, “Beziehungen der \(E_7\) und \(E_8\) zur Oktavenebene. I”, Proc. KNAW Ser. A 57 (1954), 218–230 (= Indag. Math. 16, unverified). doi:10.1016/s1385-7258(54)50032-6. Used: history.

[Tits] J. Tits, “Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I: Construction”, Proc. KNAW Ser. A 69 (1966), 223–237 (= Indag. Math. 28, unverified). doi:10.1016/s1385-7258(66)50028-2. Used: history.

[Hamilton 1844] W. R. Hamilton, “On quaternions; or on a new system of imaginaries in algebra”, London, Edinburgh and Dublin Philos. Mag. (3) 25 (1844), no. 169, 489–495. doi:10.1080/14786444408645047; open copy Zenodo 1431043. Used: the letter to Graves of 17 Oct 1843 (“occurred to me yesterday”) — quaternions. The date 16 Oct and the bridge come from Hamilton's letters of 1858 and 1865, quoted in [Baez, §1].

[Graves 1845] J. T. Graves, “On a connection between the general theory of normal couples and the theory of complete quadratic functions of two variables”, Philos. Mag. (3) 26 (1845), no. 173, 315–320. doi:10.1080/14786444508645136; open copy Zenodo 1431053. Used: P.S., p. 320 (octads “applied by me about Christmas 1843”) — octonions.

[Cayley 1845] A. Cayley, “On Jacobi's elliptic functions, in reply to the Rev. Brice Bronwin; and on quaternions”, Philos. Mag. (3) 26 (1845), no. 172, 208–211. doi:10.1080/14786444508645107; open copy Zenodo 1431049. Used: “P.S. On Quaternions”, pp. 210–211 — octonions.

[Hurwitz 1898] A. Hurwitz, “Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln”, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1898), 309–316. JFM 29.0177.01; open copy EuDML 58420. Used: p. 309, the composition identity exists only for \(n=1,2,4,8\) — 1898.

[Dickson] L. E. Dickson, “On quaternions and their generalization and the history of the eight square theorem”, Ann. of Math. (2) 20 (1919), no. 3, 155–171; addenda p. 297. doi:10.2307/1967865; open copy archive.org. Used: §4, p. 158, eq. (6) (doubling \(\Ha\to\Oc\)) — doubling; §8, p. 164 (Graves's letter of 26 Dec 1843) — octonions. Dickson doubles only \(\Ha\to\Oc\); the iterated process is later.

[GG] M. Günaydin, F. Gürsey, “Quark structure and octonions”, J. Math. Phys. 14 (1973), no. 11, 1651–1667. doi:10.1063/1.1666240. Used: abstract (split basis; subgroups \(\mathrm{SU}(3)\), \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) of \(\Gtwo\)) — physics. Full text not opened.

[GRS] F. Gürsey, P. Ramond, P. Sikivie, “A universal gauge theory model based on \(E_6\)”, Phys. Lett. B 60 (1976), no. 2, 177–180. doi:10.1016/0370-2693(76)90417-2. Used: abstract (two 27-dimensional representations) — physics.

[KT] T. Kugo, P. Townsend, “Supersymmetry and the division algebras”, Nucl. Phys. B 221 (1983), no. 2, 357–380. doi:10.1016/0550-3213(83)90584-9. Used: abstract (spinors and \(\R,\C,\Ha,\Oc\) via \(\SL(2,\K)\) for \(D=3,4,6\); \(D=10\) discussed) — physics.

[Baez–Huerta] J. C. Baez, J. Huerta, “Division algebras and supersymmetry I”, in Superstrings, Geometry, Topology, and C*-algebras, Proc. Sympos. Pure Math. 81, Amer. Math. Soc., 2010, 65–80. doi:10.1090/pspum/081/2681758; arXiv:0909.0551 (v3 checked; open copy). Used: abstract and §1, p. 1 (supersymmetric iff dimension 3, 4, 6, 10); §6 (proof of “if”) — physics.

[GST 1983] M. Günaydin, G. Sierra, P. K. Townsend, “Exceptional supergravity theories and the magic square”, Phys. Lett. B 133 (1983), 72–76. doi:10.1016/0370-2693(83)90108-9. Used: abstract — physics. Full text not opened.

[GST 1984] M. Günaydin, G. Sierra, P. K. Townsend, “The geometry of \(N=2\) Maxwell–Einstein supergravity and Jordan algebras”, Nucl. Phys. B 242 (1984), 244–268. doi:10.1016/0550-3213(84)90142-1. Used: abstract (classification by Jordan algebras; four magical cases \(\mathfrak h_3(\K)\)) — physics. Full text not opened.

[FG] S. Ferrara, M. Günaydin, “Orbits of exceptional groups, duality and BPS states in string theory”, Int. J. Mod. Phys. A 13 (1998), no. 13, 2075–2088. doi:10.1142/S0217751X98000913; arXiv:hep-th/9708025 (v2 checked; open copy). Used: abstract (cubic invariant and entropy in \(d=5\)); §2, p. 3 and eqs. (2-8)–(2-10) (split octonions, \(E_{6(6)}\), orbits) — physics; §4, pp. 9–10 (scalar manifold \(=\{\det=1\}\), orbit with three positive eigenvalues \(=G/H\), entropy \(\sqrt{|I_3|}\)) — physics, octonions over \(\C\).

[DV] M. Dubois-Violette, “Exceptional quantum geometry and particle physics”, Nucl. Phys. B 912 (2016), 426–449. doi:10.1016/j.nuclphysb.2016.04.018; arXiv:1604.01247 (v2). Used: abstract — physics.

[TDV] I. Todorov, M. Dubois-Violette, “Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra”, Int. J. Mod. Phys. A 33 (2018), no. 20, 1850118. doi:10.1142/S0217751X1850118X; arXiv:1806.09450 (v1). Used: abstract — physics. The specific subgroups were not checked.

[Boyle] L. Boyle, “The standard model, the exceptional Jordan algebra, and triality”, J. Math. Phys. 67 (2026), no. 7, 071701. doi:10.1063/5.0336175; arXiv:2006.16265 (v2). Used: abstract — physics.

[Furey] C. Furey, “\(\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y\ (\times\mathrm{U}(1)_X)\) as a symmetry of division algebraic ladder operators”, Eur. Phys. J. C 78 (2018), no. 5, 375. doi:10.1140/epjc/s10052-018-5844-7; arXiv:1806.00612 (open access). Used: abstract — physics.