is an isomorphism from our starting copy of the octonions (D(+)) to the copy altered
by the X-product (AU^{3})(U^{-3}B).
Ok, now I want to introduce some notation from my
book,
which, while not common, I find very useful.
For any octonions A and B, define
A_{L}[B] = AB;
A_{R}[B] = BA.
That is, A_{L} is the left multiplication operator, and A_{R} is the right multiplication operator.
Then the isomorphism A --> UAU^{-1} implies that the group of operators
U_{R}^{-1}U_{L}G_{2}U_{L}^{-1}U_{R}
is the automorphism group of the octonions endowed with the product (AU^{3})(U^{-3}B).
So it too is a copy of G_{2}.
This is a special case of a more general result that is on one of the following pages.
Note that if U is a 6th root of unity, then
G_{2} =
U_{R}^{-1}U_{L}G_{2}U_{L}^{-1}U_{R},
since in that case the initial product is not altered by the X-product. The calculator below
is the same one that I put on the previous page.