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  Octonions & Sedenions


It is one way of getting back from 8 dimensions to a real number via a multiplication.

  In a coordinate representation, the complex conjugate of an octonion, 
                A :=  m=0S7  am im  =  a0 i0  +  j=1S7  aj ij
may be defines as 
                A* :=  m=0S7  am im*   =  a0 i0  -  j=1S7  aj ij ,

where im  ( m = 0,....7 ) are basal elements of octonions, O, and  i0 = 1 is real (i0* =  i0  ) .  Then
                n(A) := AA* = A*A  =  a0 + j=1S7  aj2
is a real number, called the norm of A.
     In the case of complex numbers the  conjugate of  z := a + ib , is z* := a - ib , and is a reflection of z about the real number axis. It requires only a little imagination to visualize the conjugate of an octonion.
     As with complex numbers,     (A*)* = A  .
     If A and B are two octonions, then     ( A B )* = B* A*  .
 ( i.e. it is said that the conjugation is an anti automorphism ).
  One can find the inverse, A-1, of a not zero octonion, A, from its conjugate :   AA* = n (A)  implies that

n (A)
   = I  ,   which implies that  A-1   = 

The conjugation is just one case of a more general concept of involution.