Cayley-Dickson construction

In the foregoing discussion, an algebraPlanetmathPlanetmath shall mean a non-associative algebra.

Let A be a normed *-algebra, an algebra admitting an involutionPlanetmathPlanetmath ( *, over a commutative ring R with 10. The Cayley-Dickson construction is a way of enlarging A to a new algebra, KD(A), extending the * as well as the norm operations in A, such that A is a subalgebraMathworldPlanetmath of KD(A).

Define KD(A) to be the module (external) direct sumPlanetmathPlanetmath of A with itself:


Therefore, addition in KD(A) is defined by addition componentwise in each copy of A. Next, let λ be a unit in R and define three additional operations:

  1. 1.

    (Multiplication) (ab)(cd):=(ac+λd*b)(da+bc*), where * is the involution on A,

  2. 2.

    (Extended involution) (ab)*:=a*(-b), and

  3. 3.

    (Extended Norm) N(ab):=(ab)(ab)*.

One readily checks that the multiplication is bilinearPlanetmathPlanetmath, since the involution * (on A) is linear. Therefore, KD(A) is an algebra.

Furthermore, since the extended involution * is clearly bijective and linear, and that


this extended involution is well-defined and so KD(A) is in addition a *-algebra.

Finally, to see that KD(A) is a normed *-algebra, we identify A as the first componentMathworldPlanetmath of KD(A), then A becomes a subalgebra of KD(A) and elements of the form a0 can now be written simply as a. Now, the extended norm


where N in the subsequent terms of the above equation array is the norm on A given by N(a)=aa*. The fact that the N:KD(A)A, together with the equality N(00)=0 show that the extended norm N on KD(A) is well-defined. Thus, KD(A) is a normed *-algebra.

The normed *-algebra KD(A), together with the invertible element λR, is called the Cayley-Dickson algebra, KD(A,λ), obtained from A.

If A has a unity 1, then so does KD(A,λ) and its unity is 10. Furthermore, write i=01, we check that, ia=(01)(a0)=0a*=(a*0)(01)=a*i. Therefore, iA=Ai and we can identify the second component of KD(A,λ) with Ai and write elements of Ai as ai for aA.

It is not hard to see that A(Ai)=(Ai)AAi and (Ai)(Ai)A. We are now able to write


where each element xKD(A,λ) has a unique expression x=a+bi.

Properties. Let x,y,z will be general elements of KD(A,λ).

  1. 1.


  2. 2.


  3. 3.


Examples. All examples considered below have ground ring the reals .

  • KD(,-1)=, the complex numbersMathworldPlanetmathPlanetmath.

  • KD(,-1)=, the quaternions.

  • KD(,-1)=𝕆, the octonionsMathworldPlanetmath.

  • KD(𝕆,-1)=𝕊, which are called the sedenions, an algebra of dimension 16 over .


  1. 1.

    Starting from , notice each stage of Cayley-Dickson construction produces a new algebra that loses some intrinsic properties of the previous one: is no longer orderable (or formally real); commutativity is lost in ; associativity is gone from 𝕆; and finally, 𝕊 is not even a division algebraMathworldPlanetmath anymore!

  2. 2.

    More generally, given any field k, any algebra obtained by applying the Cayley-Dickson construction twice to k is called a quaternion algebra over k, of which is an example. In other words, a quaternion algebra has the form


    where each λik*:=k-{0}. Any algebra obtained by applying the Cayley-Dickson construction three times to k is called a Cayley algebra, of which 𝕆 is an example. In other words, a Cayley algebra has the form


    where each λik*. A Cayley algebra is an octonion algebra when λ1=λ2=λ3=-1.


Title Cayley-Dickson construction
Canonical name CayleyDicksonConstruction
Date of creation 2013-03-22 14:54:11
Last modified on 2013-03-22 14:54:11
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 25
Author CWoo (3771)
Entry type Definition
Classification msc 17A99
Synonym Cayley-Dickson process
Synonym doubling process
Synonym octonion algebra
Related topic TheoremsOnSumsOfSquares
Defines Cayley-Dickson algebra
Defines sedenion
Defines quaternion algebra
Defines Cayley algebra