Here's a more general X,Y-product example. Define

AoB = (AU

where U and V are arbitrary unit octonions. So this is a completely general

X,Y-product. Again denote the resulting octonion variant by

of G

A --> V

In particular,

AoB = U

So A --> U

from

G

Finally an even more general result, from which one may construct an even more general

result inductively. It is another example of

g

where AoB is again an X,Y-product. To save typing I'll just give g, X and Y:

g[A] = W

X = U

Y

Clearly there is a good deal of redundancy, there being many elements of SO(8) unravelled

by the same X,Y-product. The shape of the submanifold of SO(8) associated with every

pair (X,Y) in S

then the manifold is just G

and h is an element of G

hg[AoB] = h[g[A]g[B]] = hg[A]hg[B],

so G

if g and f are unravelled by the same X,Y-product, then gf

These ideas give us an explicit extension of G

SO(8) --> G

The 14-dimensional space of unit pairs (X,Y), modulo sign change, together with

the 14-dimensional G

These ideas can be applied to the quaternions and complex numbers as well. In

these two associative cases the X,Y-product reduces to what I'll call a W-product:

AoB = (AX)(Y

where W is either an arbitrary unit quaternion or complex number. The manifold of W's

will extend the respective automorphism groups to SO(4) in the quaternion case, and SO(2)

in the complex case. In the quaternion case the automorphism group is the 3-dimensional SO(3),

and the W-manifold is S

In the complex case the automorphism group is the 0-dimensional discrete group Z

the W-manifold is S

If g is an element of SO(2) or SO(4), respectively, then there is some unit W such that

the product rule still holds in an associative form:

g[AWB] = g[A]g[B].

Clearly this is true, for example, if g[A] = WA, which is an SO(2) action if W is complex,

and an SO(4) action if W is quaternion.

This may finish this section looking at the relationship of SO(8) to the X,Y-product.

First two useful identities:

g

T

Actually, since these equations are 0 --> 1 --> 2 --> 0 invariant, this is six useful identities.

These imply

g

This is a fairly useful result. If g

then g

And if g

then g

For example, if X = U

V

so

g

and

g

If we set g

g

as we mentioned previously. This gives us a nice way to generate other elements of

SO(8) to unravel X,Y-products.

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