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  Octonions & Sedenions

 
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Need Some Identities - the algebraic ones ?

     Identities are routinely used to simplify or to reveal some underlying structures of mathematical expressions. Because of the non-associative nature of octonions the non-trivial identities involving three or more octonions draw greater interests. Here is a list of few identities for octonions.

     Let A, B, C, ...X, Y, represent octonions. 

(A, B, C)  =  - (A, C, B)  =   (B, C,A)  ,.... etc.,
         i.e.,  skew symmetric in its three variables.

(X, X, A)  =  0  =  (X, A, X)  ,....  etc.,
        i.e., vanishes whenever two variables are same.

(X, X*, A)  =  0  =  (X, X -1, A)  ,  where X*, and X -1 are conjugate and inverse of X, respectively.

(AB, C)  -  A (B, C)  -  (A, C) B  =  3 (A, B, C)

The next three identities are collectively known as Moufang identities:

(XA, X, B)  +  (X, A, XB)  =  0 ,     or    ( ( XA) X) B  -  X (A(XB) ) =  0

(B, XA, X)  =  - (B, X, A) X ,        or    B ( (XA) X )  =  ( (BX) A ) X

(XA) (BX)  =  X (AB) X
 

(X 2 , A, B)  =  X (X, A, B)  +  (X, A, B) X

    ( (X, Y, A) , X, Y)  =  -  (X, Y, A) (X, Y)  =  (X, Y) (X, Y, A)

    ( (X, Y, A)2 , X, Y)  =   0  ,   Associative law holds for some!

     ( (X, Y, A)2 , A)  =  0  =  ( (X, Y, A)4 , A)  =  0  ,     Commutative law holds for some!

 

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