The fundamentals of cubic norm structures and Jordan algebras are laid out in the first two sections of the chapter. We then derive elementary principles of building up “big” cubic norm structures out of “smaller” ones before we proceed to study cubic Jordan matrix algebras, the most important “hands-on” examples of cubic Jordan algebras. Next we turn to elementary idempotents, which will be used to present a special version of the Jacobson Coordinatization Theorem. Proceeding to Freudenthal algebras, we show that they exist only in ranks 1, 3, 6, 9, 15, and 27, with those of rank 27 being (finally!) called Albert algebras. We define the notion of a split Freudenthal algebra and prove, in analogy with composition algebras, that all Freudenthal algebras are split by some faithfully flat extension, though not always by an étale cover. After having investigated isotopies and norm similarities, with an important characterization of isotopes in Jordan matrix algebras over LG rings as its central result, we study reduced Freudenthal algebras over fields by exhibiting various classifying quadratic form invariants, particularly those pertaining to the invariants mod 2 of Albert algebras.

This chapter is devoted to a short excursion into the theory of alternative algebras over commutative rings. We prove Artin’s theorem, according to which alternative algebras on two generators are associative. We study in detail McCrimmon’s theory of homotopes, which plays a strong role in this book and therefore merits greater emphasis here than one might find in other discussions of alternative algebras.

This chapter provides an in-depth study of composition algebras over commutative rings, which we carry out in the more general framework of conic algebras (called quadratic algebras or algebras of degree 2 by other authors). We present the Cayley–Dickson construction and define composition algebras as unital nonassociative algebras that are projective as modules and allow a non-singular quadratic form permitting composition. We use this construction to obtain first examples of octonion algebras more general than the Graves–Cayley octonions and to derive structure theorems for arbitrary composition algebras. Specializing, it is shown that all composition algebras of rank at least 2 over an LG ring arise from an appropriate quadratic étale algebra by the Cayley–Dickson construction. Other examples of octonion algebras are obtained using ternary hermitian spaces. We address the norm equivalence problem, which asks whether composition algebras are classified by their norms and has an affirmative answer over LG rings but not in general. After a short excursion into affine (group) schemes, we conclude the chapter by showing that arbitrary composition algebras are split by étale covers.

This chapter is concerned with the two Tits constructions of cubic Jordan algebras over commutative rings, which we present for the first time in book form. The key feature of the first Tits construction is that it starts out from cubic alternative algebras rather than cubic associative ones; the key notion that keeps the construction going is that of a Kummer element. Similar statements apply to the second Tits construction, where Kummer elements are replaced by étale elements. Such elements are available in abundance over residually big LG rings. It follows that all Albert algebras over such rings or over arbitrary fields may be obtained from the second Tits construction. The chapter concludes with an application to cubic Jordan division algebras over fields. We show that they are either purely inseparable field extensions of characteristic 3 or Freudenthal algebras of dimension 1, 3, 9, or 27. In each of these dimensions, we construct examples over appropriate fields and conclude the section by showing that over the “standard” fields (the complex numbers, the real numbers, finite, local and global ones), Albert division algebras do not exist.

In most of this book, we have studied Albert and octonion algebras. In this chapter, we connect those with the theory of semi-simple affine group schemes, especially those of type E6, F4, and G2. As part of this effort, we give an introduction to non-abelian flat cohomology and its applications to descent. We leverage this together with known results about affine group schemes such as Gross’s mass formula to classify the Albert algebras over the integers, a recently discovered result.

This chapter will develop from scratch the elementary theory of (quadratic) Jordan algebras over commutative rings. After a brief account of linear Jordan algebras and their most rudimentary properties over rings in which 2 is invertible, we proceed to para-quadratic algebras, which play the same role in the quadratic setting as is played by ordinary nonassociative algebras in the linear setting. Quadratic Jordan algebras are introduced. We derive a wide range of useful identities and acquaint the reader with the standard examples of special Jordan algebras, namely the Jordan algebra constructed from a unital associative algebra, from an associative algebra with involution, or from a pointed quadratic module. After a brief interlude concerning a peculiar class of two-variable identities, we investigate what are arguably the most important concepts of the theory: invertibility, isotopy, and the structure group. The chapter concludes with a concise description of the Peirce decomposition relative to an idempotent, and also relative to a complete orthogonal system of idempotents.

The end of this book will concern connections between Freudenthal and composition algebras on the one hand and Lie algebras and group schemes on the other. We begin with Lie algebras, the subject of this chapter. The classification of finite-dimensional simple Lie algebras over the complex numbers leads to the notion of root system, a language that will be used for the rest of the book. In that classification, one finds infinite families that are related to the unitary, orthogonal and symplectic involutions of n-by-n matrices. The five isolated cases are usually referred to as exceptional, and those cases are where we find the closest links with Albert and octonion algebras. Most of this chapter is devoted to the study of the algebra of derivations of a non-associative or para-quadratic algebra.

This chapter starts out with a short introduction to the language of nonassociative algebras over commutative rings. It then proceeds to familiarize the reader with two of the most important elementary techniques utilized in this book: scalar extensions (also known as base change) and finitely generated projective modules. Standard properties of involutions and quadratic maps are also recalled before we conclude the chapter with a short introduction into Roby’s theory (1963) of polynomial laws.

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.