Quaternions & Octonions (Re: so^' sie^u phu'c)

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Hi friends,

There is a famous classical theorem of algebra which say that:

Any finite dimensional algebra over R (attention: may be non-associative)
without zero divizors can have dimension equal: 1, 2, 4 or 8.
A Example of 8-dimensional algebra which is non-associative is Cayley
algebra.  Ceyley algebra is defined by H + H*epsilon.  Here + denotes
a direct sum and H is algebra of quaternions.  Denote that there are other
8-dimensional algebras those are not isomorphic to Ceyley algebra.
But, each 8-dimensional algebra (with above assumptions) which
is alternative (i.e. x.x.y=y.x.x=0) and non-associative is isomorphic to 
Ceyley algebra.
So, in the category of 8-dim alternative and non-associative algebras,
Ceyley algebra is unique!!!

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And about associative algebra we have a beautiful Frobenius theorem which
says that: 
any finite dimensional associative algebra over R without zero divizors
are isomorphic to R (real numbers), C (complex numbers) or H 
(quaternions).

enjoys,
SN
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