do not form an associative algebra they can not be represented by matrices.
However this does not preclude us from constructing matrices with octonion
entrices or representing adjoint algebra of octonions in terms of matrices.
The following describes two methods of representing octonions.
(1) Quaternionic representation:
An octonion, A, is written as an ordered pair of two
4-dimensional quaternions, Q1 and Q2
A := ( Q1;
Q2 ) .
For the algebra of octonions, O
, to satisfy the basic properties outlined in Introduction one needs to
define the multiplication rules of two octonions, A and B.
Writing B in terms of two quaternions, Q3
and Q4, we define the multiplication of
A and B, via Cayley-Dickson process
AB := ( Q1;
Q2 )( Q3;
:= ( Q1Q3
where Q* is a conjugate of Q, and b
is a field parameter.
(2) Coordinate representation:
An octonion, A, may be written componentwise
A := m=0S7
= a0 i0
aj ij ,
are the basal elements of O. The basal
element, i0, is chosen as the basis
of reals, R. For the algebra of
octonions to satisfy the basic properties outlined in Introduction one
needs to define the multiplication rules for the basis. There are many
choices for this task but we will describe just two of them below.
(I) via congruence modulo
Define ij ij
= -1 ; ij ij+1
:= ij+3 := ij-4
, (mod 7)
i2m = i2n,
whenever ik im
= in .
Then the multiplication rule for the seven basal elements,
ij ( j = 1,...7)
is summed up in the following seven cycles:
(1,3,7) , (1,5,6) , (2,3,5) , (2,6,7) , (3,4,6)
, (4,5,7) .
The cycle (j,k,m) is to mean
im , ik
im = ij
, ik ij
= - im .
(II) via Cayley-Dickson process
from the quaternion basis:
Write octonion basal element as ordered pair of quaternion
:= ( im ; 0
) , and
( 0 ; im ),
for m = 0,1,2,3.
= ( i3 ; 0 ) ,
and i6 = ( 0 ; i2
Then by Cayley-Dickson
process the multiplication rule can be found,
for example, i3 i6=
( 0 ; i2 i3)
= ( 0 ; i1 ) = i5,
and summarized in the following seven cycles:
(1,2,3) , (1,4,5) , (1,7,6) , (2,4,6) , (2,5,7) , (3,4,7)
, (3,6,5) .