On Quaternions and Octonions

Contents
Preface
I The Complex Numbers and Their Applications to 1- and 2- Dimensional Geometry
Introduction
1.1 The Algebra R of Real Numbers
1.2 Higher Dimensions
1.3 The Orthogonal Groups
1.4 The History of Quaternions and Octonions
Complex Numbers and 2- Dimensional Geometry
2.1 Rotations and Reflections
2.2 Finite Subgroups of GO2 and SO2
2.3 The Gaussian Integers
2.4 The Kleinian Integers
2.5 The 2-Dimensional Space Groups
II The Quaternions and Their Applications to 3- and 4- Dimensional Geometry
Quaternions and 3- Dimensional Groups
3.1 The Quaternions and 3-Dimensional Rotations
3.2 Some Spherical Geometry
3.3 The Enumeration of Rotation Groups
3.4 Discussion of the Groups
3.5 The Finite Groups of Quaternions
3.6 Chiral and Achiral, Diploid and Haploid
3.7 The Projective or Elliptic Groups
3.8 The Projective Groups Tell Us All
3.9 Geometric Description of the Groups
Appendix: v --> vqv Is a Simple Rotation
Quaternions and 4- Dimensional Groups
4.1 Introduction
4.2 Two 2-to-1 Maps
4.3 Naming the Groups
4.4 Coxeter’s Notations for the Polyhedral Groups
4.5 Previous Enumerations
4.6 A Note on Chirality
Appendix: Completeness of the Tables
The Completeness of Tables 4.1 and 4.2
The Completeness of Table 4.3
The Last Eight Lines of Table 4.3
The Hurwitz Integral Quaternions
5.1 The Hurwitz Integral Quaternions
5.2 Primes and Units
5.3 Quaternionic Factorization of Ordinary Primes
5.4 The Metacommutation Problem
5.5 Factoring the Lipschitz Integers
5.5.1 Counting Lipschitzian Factorizations
III The Octonions and Their Applications to 7- and 8- Dimensional Geometry
The Composition Algebras
6.1 The Multiplication Laws
6.3 The Doubling Laws
6.4 Completing Hurwitz’s Theorem
6.5 Other Properties of the Algebras
6.6 The Maps Lx, Rx, and Bx
6.7 Coordinates for the Quaternions and Octonions
6.8 Symmetries of the Octonions: Diassociativity
6.9 The Algebras over Other Fields
6.10 The 1-, 2-, 4-, and 8-Square Identities
6.11 Higher Square Identities: Pfister Theory
Appendix: What Fixes a Quaternion Subalgebra?
Moufang Loops
7.1 Inverse Loops
7.2 Isotopies
7.3 Monotopies and Their Companions
7.4 Different Forms of the Moufang Laws
Octonions and 8- Dimensional Geometry
8.1 Isotopies and SO8
8.2 Orthogonal Isotopies and the Spin Group
8.3 Triality
8.4 Seven Rights Can Make a Left
8.5 Other Multiplication Theorems
8.6 Three 7-Dimensional Groups in an 8-Dimensional One
8.7 On Companions
The Octavian Integers O
9.1 Defining Integrality
9.2 Toward the Octavian Integers
9.3 The E8 Lattice of Korkine, Zolotarev, and Gosset
9.3.1 The Simplex Lattice An
9.3.2 The Orthoplex Lattice Dn
9.3.3 Defining E8
9.4 Division with Remainder, and Ideals
9.5 Factorization in O^8
9.5.1 The Structure of the Divisor Sets
9.6 The Number of Prime Factorizations
9.7 “Meta-Problems” for Octavian Factorization
Automorphisms and Subrings of O
10.1 The 240 Octavian Units
10.2 Two Kinds of Orthogonality
10.3 The Automorphism Group of O
10.4 The Octavian Unit Rings
10.5 Stabilizing the Unit Subrings
10.5.1 Subring G^{4−}
10.5.2 Subring H^4
10.5.3 Subring E^4
Appendix: Proof of Theorem 5
Reading O Mod 2
11.1 Why Read Mod 2?
11.2 The E8 Lattice, Mod 2
11.3 What Fixes <λ>?
11.3.1 The Calculations
11.4 The Remaining Subrings Modulo 2
The Octonion Projective
Plane OP2
12.1 The Exceptional Lie Groups and Freudenthal’s “Magic Square”
12.2 The Octonion Projective Plane
12.3 Coordinates for OP^2
Bibliography
Index