Baez E7 and exceptional Jordan-algebra paper expansion

Core idea

This permanent provenance and reading index records 125 new concept knowls, the new nonassociative-algebra subject page, and one expanded existing entry. The expansion follows John Baez's Three Generations in E7 and John C. Baez and Paul Schwahn's The Standard Model Gauge Group from the Exceptional Jordan Algebra. The order below starts with reusable prerequisites and ends with the papers' principal constructions and theorems.

Lie-algebra and root-system foundations

• Rank of a semisimple Lie algebra
• Regular Lie subalgebra
• Compatible Cartan subalgebras
• Root subsystem
• Simply laced root system
• Type $A$ root system
• Type $D$ root system
• Exceptional $E$-type root systems
• Root $\mathfrak{sl}_2$-subalgebra
• Compact real form
• Levi subalgebra
• Maximal Levi subalgebra
• Removing a simple root
• Maximal-rank Lie subalgebra
• Maximal Lie subalgebra
• Centralizer of a Lie subalgebra
• Centralizer of a regular reductive subalgebra
• Mutual centralizers in a Lie algebra
• Borel–de Siebenthal theory
• Identity component of a Lie group
• Transitivity on real Grassmannians

Named Lie algebras and exceptional groups

• Complex Lie algebra $\mathfrak{sl}_3(\mathbb C)$
• Complex Lie algebra $\mathfrak{sl}_5(\mathbb C)$
• Complex Lie algebra $\mathfrak{sl}_6(\mathbb C)$
• Complex Lie algebra $\mathfrak{so}_{10}(\mathbb C)$
• Complex Lie algebra $\mathfrak{so}_{12}(\mathbb C)$
• Exceptional Lie algebra $\mathfrak e_6$
• Exceptional Lie algebra $\mathfrak e_7$
• Exceptional Lie algebra $\mathfrak e_8$
• Exceptional Lie algebra $\mathfrak f_4$
• Exceptional Lie algebra $\mathfrak g_2$
• Compact exceptional Lie group $E_6$
• Compact exceptional Lie group $E_7$
• Compact exceptional Lie group $E_8$
• Compact exceptional Lie group $F_4$
• Compact exceptional Lie group $G_2$
• The $E_n$ series of Lie algebras

Representation-theoretic tools

• Branching rule for Lie representations
• Defining representation of a classical Lie algebra
• Exterior-power representation
• Even and odd exterior algebra
• Exterior algebra of a direct sum
• Fixed-vector subspace
• Multiplicity of an irreducible representation
• Half-spin representation
• $\operatorname{Spin}(8)$ triality
• The isomorphism $\operatorname{Spin}(6)\cong SU(4)$
• Dynkin-diagram automorphisms and dual representations
• Root lattice
• Weight lattice
• Weights of the defining $\mathfrak{sl}_3$-representation

The three-generation construction in E7

• Good Standard Model embedding in $\mathfrak e_7$
• Exceptional chain to the Standard Model Lie algebra
• Generation $\mathfrak{sl}_3$ in $\mathfrak e_7$
• Generation plane in $\mathfrak e_7$
• Generation $\mathfrak{sl}_2$-subalgebras
• $E_7$ root-projection trichotomy
• Three $D_6$ subsystems in $E_7$
• $E_7$ branching under $A_1+D_6$
• Standard $\mathfrak{sl}_6$ in $\mathfrak e_7$
• Intersection of the three $A_1+D_6$ subalgebras
• Standard $\mathfrak{sl}_5$ in $\mathfrak e_7$
• Thirty-dimensional generation module in $\mathfrak e_7$
• $E_7$ branching under $A_2+A_5$
• Generation module as an even exterior algebra
• Three-generation decomposition of $\mathfrak e_7$

Standard Model bridge and quantum terminology

• Central quotient of a Lie group
• Maximal connected closed subgroup
• Special block unitary group
• Standard Model gauge group
• Standard Model Lie algebra
• Georgi–Glashow $SU(5)$ embedding
• Standard Model exterior-algebra representation
• Standard Model fermion generation
• Right-handed neutrino gauge singlet
• Weak hypercharge
• Standard Model gauge group as an $F_4$ stabilizer intersection
• Qubit
• Qutrit
• Antiunitary operator

Nonassociative, composition, and octonion algebras

• Nonassociative algebra
• Alternative algebra
• Composition algebra
• Real normed division algebra
• Hurwitz's theorem on normed division algebras
• Octonion algebra
• Octonion conjugation, norm, and inner product
• Octonions as complex vectors
• Conjugated cross product on $\mathbb C^3$
• $SU(3)$ stabilizer of a complex octonion subalgebra

Jordan-algebra foundations and examples

• Jordan algebra
• Jordan algebra homomorphism
• Jordan subalgebra
• Ideal in a Jordan algebra
• Power-associative algebra
• Special and exceptional Jordan algebras
• Euclidean Jordan algebra
• Simple Euclidean Jordan algebra
• Hermitian matrix Jordan algebra
• Spin-factor Jordan algebra
• Complex-qubit Jordan algebra
• Complex-qutrit Jordan algebra
• Octonionic spin factor
• Exceptional Jordan algebra
• Automorphism group of a Jordan algebra
• Derivation of a Jordan algebra
• Trace form of a Euclidean Jordan algebra
• Spectral theorem for Euclidean Jordan algebras
• Automorphism transitivity on Jordan frames

Peirce theory and exceptional stabilizers

• Jordan idempotent
• Orthogonal Jordan idempotents
• Primitive Jordan idempotent
• Jordan frame
• Peirce decomposition of a Jordan algebra
• Peirce-one Jordan corner
• Frame decomposition of Hermitian Jordan algebras
• Unique octonionic spin-factor corner
• Complex-subalgebra decomposition in the Albert algebra
• $\operatorname{Spin}(8)$ stabilizer of an Albert-algebra frame
• $\operatorname{Spin}(9)$ stabilizer of an octonionic spin factor
• Complex-qutrit stabilizer in $F_4$
• $F_4$ transitivity on complex-qutrit subalgebras
• $F_4$ transitivity on compatible Jordan-subalgebra pairs

Jordan triple systems

• Hermitian Jordan triple system
• Tripotent in a Jordan triple system

Expanded existing entry

• $SL(2,\mathbb C)$ and $\mathfrak{sl}_2(\mathbb C)$

References

1. John C. Baez, “Three Generations in E7,” 2026. arXiv:2608.06271.
2. John C. Baez and Paul Schwahn, “The Standard Model Gauge Group from the Exceptional Jordan Algebra,” 2026. arXiv:2606.15235.