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An octonion form an 8-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the quaternions. . . . . . .


Communtative: No.

Associative: No.

Degen's Eight Square Identity

(a²+b²+c²+d²+e²+f²+g²+h²)(m²+n²+o²+p²+q²+r²+s²+t²) =
   (am-bn-co-dp-eq-fr-gs-ht)² +
   (bm+an+do-cp+fq-er-hs+gt)² +
   (cm-dn+ao+bp+gq+hr-es-ft)² +
   (dm+cn-bo+ap+hq-gr+fs-et)² +
   (em-fn-go-hp+aq+br+cs+dt)² +
   (fm+en-ho+gp-bq+ar-ds+ct)² +
   (gm+hn+eo-fp-cq+dr+as-bt)² +

The identity follows from the fact that the norm of the product of two octonions is the product of the norms, in a way similar to the quaternion identity and the complex product identity (see Sums of Squares).

Multiplication table

b.1 b.e1 b.e2 b.e3 b.e4 b.e5 b.e6 b.e7
a.1 1 e1 e2 e3 e4 e5 e6 e7
a.e1 e1 -1 e4 e7 -e2 e6 -e5 -e3
a.e2 e2 -e4 -1 e5 e1 -e3 e7 -e6
a.e3 e3 -e7 -e5 -1 e6 e2 -e4 e1
a.e4 e4 e2 -e1 -e6 -1 e7 e3 -e5
a.e5 e5 -e6 e3 -e2 -e7 -1 e1 e4
a.e6 e6 e5 -e7 e4 -e3 -e1 -1 e2
a.e7 e7 e3 e6 -e1 e5 -e4 -e2 -1

 fix this table.


Fano plane

 . . . . . . explain how fano plane works for multiplication.  Add the fano plane to the geometry glossary.

Internet references

Wikipedia: Octonion

Mathworld: Octonion and Degen's Eight-Square Identity

Euclidean Space: octonion

Fact Archive: Octonion

Tony Smith:

. . . . . . relate this to 3x3 matrices

Related pages in this website

Complex, Quaternion, Octonion, and Sedenion numbers are n-tuples of real numbers, where n=2,4,8,16, respectively.

Sums of Squares - Every positive integer can be expressed as the sum of four squares; Primes of the form 4k+1 (and their products using the Complex Product Identity) can be expressed as the sum of two squares; Primes of the form 4k+3 (and their products using the Quaternion Identity) can be expressed as the sum of four squares.

The webmaster and author of this Math Help site is Graeme McRae.