I - Prologue: the ancient protagonists

Summary

This introductory chapter provides a first glimpse at the principal characters of the book: Albert algebras and octonions, but also at other members of the families of Freudenthal and composition algebras. They are presented here in the familiar surroundings of the field of real numbers and over the integers. This has the advantage of following rather closely the historical development of the subject (stretching back into the nineteenth century), and of providing a first motivation for the study of quadratic Jordan algebras. Following Zorn (1933), we define the algebra of Graves–Cayley octonions, allowing us to view the Hamiltonian quaternions as an appropriate subalgebra. We describe in detail the Z-algebras of Hurwitz quaternions (1896) and of Dickson–Coxeter octonions (1923, 1946), which in turn give rise to our first encounter with Albert algebras over the integers.