Every unital composition algebra over a field K can be obtained by repeated application of the Cayley–Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(K) = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8.
1-dimensional composition algebras only exist when char(K) ≠ 2.
Composition algebras of dimension 1 and 2 are commutative and associative.
The squaring functionN(x) = x2 on the real number field forms the primordial composition algebra.
When the field K is taken to be real numbers R, then there are just six other real composition algebras.: 166
In two, four, and eight dimensions there are both a division algebra and a "split algebra":
binarions: complex numbers with quadratic form x2 + y2 and split-complex numbers with quadratic form x2 − y2,
Historically, the first non-associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras...: 61
In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unite, and for quaternions q and Q writes a Cayley number q + Qe. Denoting the quaternion conjugate by q′, the product of two Cayley numbers is
The conjugate of a Cayley number is q' – Qe, and the quadratic form is qq′ + QQ′, obtained by multiplying the number by its conjugate. The doubling method has come to be called the Cayley–Dickson construction.
^ abGuy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society, ISBN 978-0-8218-4459-5