Publisher Summary This chapter focuses on composition algebras, which are a natural generalization of the algebras of complex numbers, quaternions, and Cayley numbers, and which play an important role in the theory of alternative and Jordan algebras. It demonstrates that with the exception of some algebras, each simple quadratic alternative algebra is a composition algebra. The identities defining the variety of alternative algebras are written in the form (x, x, y ) = 0, and (x, y, y) = 0. The first of these identities is called the left alternative identity, and the second is the right alternative identity. Linearizing the left and right alternative identities leads to identities from which it follows that in an alternative algebra, the associator is a skew-symmetric function of its arguments.