Let be the quaternionsMathworldPlanetmath over the reals . Apply the Cayley-Dickson construction to once, and we obtain an algebraMathworldPlanetmathPlanetmathPlanetmath, variously called Cayley algebra, the octonion algebra, or simply the octonions, over . Specifically the construction is carried out as follows:

  1. 1.

    Form the vector space 𝕆=𝐤; any element of 𝕆 can be written as a+b𝐤, where a,b;

  2. 2.

    Define a binary operationMathworldPlanetmath on 𝕆 called the multiplication in O by


    where a,b,c,d, and c¯ is the quaternionic conjugationMathworldPlanetmath of c. When b=d=0, the multiplication is reduced the multiplication in . In addition, the multiplication rule above imply the following:

    a(d𝐤)=(da)𝐤 (1)
    (b𝐤)c=(bc¯)𝐤 (2)
    (b𝐤)(d𝐤)=-d¯b. (3)

    In particular, in the last equation, if b=d=1, 𝐤2=-1.

  3. 3.

    Define a unary operation on 𝕆 called the octonionic conjugation in O by


    where a,b. Clearly, the octonionic conjugation is an involutionPlanetmathPlanetmath (http://planetmath.org/Involution2) (x¯¯=x).

  4. 4.

    Finally, define a unary operation N on 𝕆 called the norm in O by N(x):=xx¯, where x𝕆. Write x=a+b𝐤, then


    It is not hard to see that N(x)=0 iff x=0.

The above four (actually, only the first two suffice) steps makes 𝕆 into an 8-dimensional algebra over such that is embedded as a subalgebraMathworldPlanetmathPlanetmathPlanetmath.

With the last two steps, one can define the inverseMathworldPlanetmathPlanetmathPlanetmath of a non-zero element x𝕆 by


so that xx-1=x-1x=1. Since x is arbitrary, 𝕆 has no zero divisorsMathworldPlanetmath. Upon checking that x-1(xy)=y=(yx)x-1, the non-associative algebra 𝕆 is turned into a division algebraMathworldPlanetmath.

Since N(x)0 for any x𝕆, we can define a non-negative real-valued function on 𝕆 by x=N(x). This is clearly well-defined and x=0 iff x=0. In addition, it is not hard to see that, for any r and x𝕆, rx=|r|x, and that satisfies the triangular inequality. This makes 𝕆 into a normed division algebra.

Since the multiplication in is noncommutative, 𝕆 is noncommutative. In fact, if we write =𝐣, where are the complex numbers and 𝐣2=-1, then B={1,𝐢,𝐣,𝐢𝐣} is a basis for the vector space over . With the introduction of 𝐤𝕆, we quickly check that 𝐤 anti-commute with the non-real basis elements in B:


Furthermore, one checks that 𝐢(𝐣𝐤)=(𝐣𝐢)𝐤=-(𝐢𝐣)𝐤, so that 𝕆 is not associative.

Since 𝕆=𝐤, the set {1,𝐢,𝐣,𝐢𝐣,𝐤,𝐢𝐤,𝐣𝐤,(𝐢𝐣)𝐤}(=BB𝐤) is a basis for 𝕆 over . A less messy way to represent these basis elements is done the following assignment:

Title octonion
Canonical name Octonion
Date of creation 2013-03-22 15:21:42
Last modified on 2013-03-22 15:21:42
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Definition
Classification msc 17A75
Classification msc 17D05
Synonym Cayley algebra
Related topic TheoremsOnSumsOfSquares
Related topic DivisionAlgebra
Defines octonion algebra