Martin Cederwall introduced the octonion X-product a few years ago
(it appeared first in Nuc Phys B). Let X be a unit octonion, then whatever
product rule you start with, the new product defined by
A o_{x} B = (AX)(X^{*}B)
is also a valid octonion product (X^{*} is the conjugate of X, and although XX^{*}=1,
because of nonassociativity (AX)(X^{*}B) is not in general equal AB), in this case leaving
the identity invariant (the X,Y-product, which generalizes the X-product,
can change the role of the unit e_{0}).
This element and it's conjugate are very special in that they are invariant under index
doubling and cycling (they also appear as essential components of my own, and Dick Gross's,
representations of the Leech lattice based on the octonions - two representations both of
which require X_{0}).
Anyway, this element also can be used to interchange the D(+) and C(+) products. Let
A and B be arbitrary octonions, and let AB be the D(+) product of A and B. Then
(AX_{0})(X_{0}^{*}B)
is the C(+) product of A and B. In particular, given
e_{a} e_{a+1} = e_{a+5},
then
(e_{a}X_{0}) (X_{0}^{*}e_{a+1}) = e_{a+3}.
All of this can be checked using the X-product calculator below, where X is set equal to X_{0}.
The A and B fields are input fields. Make sure that each of the 16 component fields has a number
in it, for I didn't bother to have the program check for this.
By the way, because of associativity the complex and quaternion algebras have no X-product variants
(although they do have X,Y-product variants).