Cayley-Dickson construction

[Joerg Arndt]

... for hypercomplex numbers.

corresponding sequence(s) do not seem exist.
There is only id:A096809 which could use much clarification.

With the CD construction I obtain for the octonions
(a number N shall denote the N-th unit EN):

           0   1   2   3   4   5   6   7 

    0:    +0  +1  +2  +3  +4  +5  +6  +7 
    1:    +1  -0  -3  +2  -5  +4  +7  -6 
    2:    +2  +3  -0  -1  -6  -7  +4  +5 
    3:    +3  -2  +1  -0  -7  +6  -5  +4 
    4:    +4  +5  +6  +7  -0  -1  -2  -3 
    5:    +5  -4  +7  -6  +1  -0  +3  -2 
    6:    +6  -7  -4  +5  +2  -3  -0  +1 
    7:    +7  +6  -5  -4  +3  +2  -1  -0 


In general, EN*EM = EH where H = N XOR M

For the the signs, (16x16 case):

             0 1 2 3 4 5 6 7  8 9 a b c d e f

       0:    + + + + + + + +  + + + + + + + +
       1:    + - - + - + + -  - + + - + - - +
       2:    + + - - - - + +  - - + + + + - -
       3:    + - + - - + - +  - + - + + - + -
       4:    + + + + - - - -  - - - - + + + +
       5:    + - + - + - + -  - + - + - + - +
       6:    + - - + + - - +  - + + - - + + -
       7:    + + - - + + - -  - - + + - - + +

       8:    + + + + + + + +  - - - - - - - -
       9:    + - + - + - - +  + - + - + - - +
       a:    + - - + + + - -  + - - + + + - -
       b:    + + - - + - + -  + + - - + - + -
       c:    + - - - - + + +  + - - - - + + +
       d:    + + - + - - - +  + + - + - - - +
       e:    + + + - - + - -  + + + - - + - -
       f:    + - + + - - + -  + - + + - - + -

As a matrix it is a Hadamard matrix (note that id:A096809
does not correspond to a Hadamard matrix).
The second row is the Thue-Morse sequence.
The 2**k -th rows are stretched versions of it.
(I do not see the general scheme for the not- power-of-2 rows).

There is a (IMHO beautiful!) routine for the signs,
see http://www.jjj.de/fxt/#fxtbook
section 20.6.1 "The Cayley-Dickson construction", p.525.

A slight modification gives the transpose of of the sign matrix.

Shall I enter the sign matrix (by antidiagonals)?


The variant (for the octonions) sometimes given is

             0  1  2  3  4  5  6  7

        0:  +0 +1 +2 +3 +4 +5 +6 +7            + + + + + + + +
        1:  +1 -0 +6 +4 -3 +7 -2 -5            + - + + - + - -
        2:  +2 -6 -0 +7 +5 -4 +1 -3            + - - + + - + -
        3:  +3 -4 -7 -0 +1 +6 -5 +2            + - - - + + - +
        4:  +4 +3 -5 -1 -0 +2 +7 -6            + + - - - + + -
        5:  +5 -7 +4 -6 -2 -0 +3 +1            + - + - - - + +
        6:  +6 +2 -1 +5 -7 -3 -0 +4            + + - + - - - +
        7:  +7 +5 +3 -2 +6 -1 -4 -0            + + + - + - - -

Should be entered by someone who has a better understanding
of the underlying constrction than me.