where each of the triples ,
, , , , , behaves like the quaternions. Octonions are notassociative.
They have been used in the study of eight-dimensional space, in which a general rotation
can be written as
The multiplication of the seven imaginary basis units can be encoded by the oriented Fano plane, often drawn as an octonion star. For each
oriented line
in the diagram, ;
reversing the order changes the sign. Together with , these rules determine the complete multiplication
table.
,
consists of the elements in a Cayley algebra. A
typical octonion is of the form
,
, 
, 
, 
, 
, 
behaves like the quaternions 
. Octonions are not associative.
They have been used in the study of eight-dimensional space, in which a general rotation
can be written as
in the diagram, 
;
reversing the order changes the sign. Together with 
, these rules determine the complete multiplication
table.
