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Basic Algebra:
An
octonion is a hypercomplex number which can be written as a linear combination
of eight basal elements. A set of octonions forms an algebra.
If A, B, C represent three octonions, the algebra of octonions, denoted
by O is

Non commutative : AB is not necessarily equal
to BA

Non associative : (AB)C is
not necessarily equal to A(BC)

Alternative
: A(AB) = A^{2 }B

Power associative : A^{n}
A^{m} = A^{n+m}
And, octonions form a

Composition algebra : n(AB) = n(A) n(B) ,
where
the norm of A , denoted n(A), is defined as
n(A) = AA* = A*A , A* is a conjugate of A
Division algebra : For a not zero octonion,
A, the multiplicative inverse A^{1} is also
an octonion.

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