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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