Familles d`algèbres de quaternions et d`octonions

C
R( + ) = ( + )( −) = +
( + ) ( + ) = ( −) + ( + ) .
H R
7→ σ( )
H=R⊕R.⊕R.⊕R.
= = −; = = −.

H=C⊕C=R⊕R.⊕R.⊕R.
σ( + + + ) = − − −
( ) = σ( ) = + + + R
H
R
R H R
R
σ( ) = σ( ) σ( )
( ) = ( ) ( ).

O H
H C
O=H⊕H
(,).(0,0) = .0−0. σ( ) , σ( ) .0+0.
, , 0,0∈H
(,)
(.).=.(.)

O=H⊕Hτ(,) = (σ( ),−)
O(,)=( ,). τ(,) = ( ) + ( ).
(.) = ( ) ( )
,∈O