The Spinorial Chessboard

1. Introduction.- 1.1 A Little History.- 1.2 Null Elements and Simple Spinors.- 1.3 About the Present Work.- 1.4 Motivation and Outlook.- 2. Notation and Terminology.- 3. Vector Spaces and Inner Products.- 3.1 Complex Structure in a Real Vector Space.- 3.2 Quaternionic Structure in a Real Vector Space.- 3.3 Complex Conjugation and Hermitean Forms.- 3.4 Real and Quaternionic Structures in a Complex Vector Space.- 3.5 Inner Products in Vector Spaces.- 4. Algebras and Their Representations.- 4.1 Definitions.- 4.2 Simple Algebras.- 4.3 Antiautomorphisms and Inner Products.- 4.4 Real Algebras.- 4.5 Graded Algebras.- 5. General Properties of Clifford Algebras.- 5.1 Definition and General Properties of Clifford Algebras.- 5.2 The Vector Space Structure of Clifford Algebras.- 5.3 The Graded Structure of Clifford Algebras.- 5.4 The Volume Element and Hodge Duality.- 5.5 Relation Between the Clifford Algebras of Vector Spaces of Adjacent Dimension.- 6. Complex Clifford Algebras.- 6.1 Dirac and Weyl Spinors.- 6.2 The Inner Products.- 6.3 Extensions of Representations.- 6.4 The Wall Groups.- 7. Real Clifford Algebras.- 7.1 The Index Periodicity.- 7.2 Charge Conjugation and Majorana Spinors.- 7.3 The Dirac Forms.- 7.4 Clifford Algebras of Euclidean Spaces.- 7.5 The Spinorial Chessboard.- 7.6 Summary.- References.