Let be the quaternions over the reals . Apply the Cayley-Dickson construction to once, and we obtain an algebra, variously called Cayley algebra, the octonion algebra, or simply the octonions, over . Specifically the construction is carried out as follows:
Form the vector space ; any element of can be written as , where ;
Define a unary operation on called the octonionic conjugation in by
where . Clearly, the octonionic conjugation is an involution (http://planetmath.org/Involution2) ().
Finally, define a unary operation on called the norm in by , where . Write , then
It is not hard to see that iff .
The above four (actually, only the first two suffice) steps makes
into an -dimensional algebra over such that is
embedded as a subalgebra.
With the last two steps, one can define the inverse of a non-zero element by
. Since is arbitrary, has no zero
divisors. Upon checking that , the non-associative algebra is turned into a division algebra.
Since for any , we can define a non-negative real-valued function on by . This is clearly well-defined and iff . In addition, it is not hard to see that, for any and , , and that satisfies the triangular inequality. This makes into a normed division algebra.
Since the multiplication in is noncommutative, is noncommutative. In fact, if we write , where are the complex numbers and , then is a basis for the vector space over . With the introduction of , we quickly check that anti-commute with the non-real basis elements in :
Furthermore, one checks that , so that
is not associative.
Since , the set () is a basis for over . A less messy way to represent these basis elements is done the following assignment:
|Date of creation||2013-03-22 15:21:42|
|Last modified on||2013-03-22 15:21:42|
|Last modified by||CWoo (3771)|