|Octonions & Sedenions|
|Basis for Octonions:
To construct basal elements of octonion from that of quaternions, let
ir ( r = 0,1,2,3 ) denote quaternion basis, and
em ( m = 0,1,...,7 ) denote octonion basis.
Let em := ( im ; 0 ) , for m = 0,1,...,3, and
em := ( 0 ; im-4 ) , for m = 4,...,7.
e.g. e0 = ( i0 ; 0 ) = ( 1 ; 0 ) = 1,
e2 = ( i2 ; 0 ) , e7 = ( 0 ; i3 ) .
Then the multiplication rule for the octonion basal elements can be found using Cayley-Dickson process (when m is chosen as -1 at each iteration). For example,
e4 e5 = ( 0 ; 1 ) ( 0 ; i1 ) = ( - i1* ; 0 ) = ( i1 ; 0 ) = e1 .
A multiplication table can then be
The multiplication rule for the pure
octonion basis ( ej , j = 1,2,...,7 ) can
be represented diagrammatically by
where the arrows indicate the signs of the resulting basal
Another more text friendly way to express
the multiplication rule is by means of octonion cycles. The
multiplication of 2 octonion basal elements, ej
and ek , is given by
and is given by
and as usual