In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These examples are useful composition algebras frequently applied in mathematical physics.

The Cayley–Dickson construction defines a new algebra similar to the direct sum of an algebra with itself, with multiplication defined in a specific way (different from the multiplication provided by the genuine direct sum) and an involution known as conjugation. The product of an element and its conjugate (or sometimes the square root of this product) is called the norm.

The symmetries of the real field disappear as the Cayley–Dickson construction is repeatedly applied: first losing order, then commutativity of multiplication, associativity of multiplication, and next alternativity.

More generally, the Cayley–Dickson construction takes any algebra with involution to another algebra with involution of twice the dimension.^{[1] []}

The Hurwitz's theorem (composition algebras) states that the reals, complex numbers, quaternions, and octonions are the only (normed) division algebras (over the real numbers).

The complex numbers can be written as ordered pairs (a, b) of real numbers a and b, with the addition operator being component-by-component and with multiplication defined by

A complex number whose second component is zero is associated with a real number: the complex number (a, 0) is the real number a.

The complex conjugate (a, b)* of (a, b) is given by

The conjugate has the property that

which is a non-negative real number. In this way, conjugation defines a norm, making the complex numbers a normed vector space over the real numbers: the norm of a complex number z is

Furthermore, for any non-zero complex number z, conjugation gives a multiplicative inverse,

As a complex number consists of two independent real numbers, they form a 2-dimensional vector space over the real numbers.

Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.

The next step in the construction is to generalize the multiplication and conjugation operations.

Slight variations on this formula are possible; the resulting constructions will yield structures identical up to the signs of bases.

The order of the factors seems odd now, but will be important in the next step.

The product of a nonzero element with its conjugate is a non-negative real number:

As before, the conjugate thus yields a norm and an inverse for any such ordered pair. So in the sense we explained above, these pairs constitute an algebra something like the real numbers. They are the quaternions, named by Hamilton in 1843.

As a quaternion consists of two independent complex numbers, they form a 4-dimensional vector space over the real numbers.

All the steps to create further algebras are the same from octonions on.

For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element.

This algebra was discovered by John T. Graves in 1843, and is called the octonions or the "Cayley numbers".

As an octonion consists of two independent quaternions, they form an 8-dimensional vector space over the real numbers.

For the reason of this non-associativity, octonions have no matrix representation.

The Cayley–Dickson construction can be carried on ad infinitum, at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. All the algebras generated in this way over a field are quadratic: that is, each element satisfies a quadratic equation with coefficients from the field.^{[1] []}

In 1954 R. D. Schafer examined the algebras generated by the Cayley-Dickson process over a field F and showed they satisfy the flexible identity.^[2] He also proved that any derivation algebra of a Cayley-Dickson algebra is isomorphic to the derivation algebra of Cayley numbers, a 14-dimensional Lie algebra over F.

The Cayley–Dickson construction, starting from the real numbers ℝ, generates division composition algebras. There are also composition algebras with isotropic quadratic forms that are obtained through a slight modification, by replacing the minus sign in the definition of the product of ordered pairs with a plus sign, as follows:

When this modified construction is applied to ℝ, one obtains the split-complex numbers, which are ring-isomorphic to the direct sum ℝ ⊕ ℝ (also written 2ℝ); following that, one obtains the split-quaternions, isomorphic to M2(ℝ); and the split-octonions, which are isomorphic to Zorn(ℝ). Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.^[3]

Albert (1942, p. 171) gave a slight generalization, defining the product and involution on B=A⊕A for A an algebra with involution (with (xy)* = yx) to be

for γ an additive map that commutes with * and left and right multiplication by any element. (Over the reals all choices of γ are equivalent to −1, 0 or 1.) In this construction, A is an algebra with involution, meaning:

The algebra B=A⊕A produced by the Cayley–Dickson construction is also an algebra with involution.

B inherits properties from A unchanged as follows.

Other properties of A only induce weaker properties of B: