Page 11.

Unravelling Triality with the X,Y-Product: Part 2.

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

AoB = (AU3)((U-1V3U)B),

where U and V are arbitrary unit octonions. So this is a completely general
X,Y-product. Again denote the resulting octonion variant by O~, and its version
of G2 by G2~. This X,Y-Product unravels the SO(8) action (among others, as we will see)

A --> VL2VRULUR2[A] = V2(UAU2)V.

In particular,

AoB = U-1{V-2({V2 (UAU2)V}{V2(UBU2)V}) V-1}U-2.

So A --> U-1(V-2AV-1)U-2 defines an isomorphism
from O to O~ (which implies the identity of O~ is U-1V-3U-2). And

G2~ = UR-2UL-1VR-1VL-2 G2 VL2VRULUR2.

Finally an even more general result, from which one may construct an even more general
result inductively. It is another example of

g-1[g[A]g[B]] = AoB,

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

g[A] = Wn{Vk(UpAUq)Vj}Wm;
X = Up(VjW2m-nVj-k)Uq-p;
Y* = Up-q(Vk-jW2n-mVk)Up.

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 S7xS7 (two 7-spheres) is easily determined. If X = Y = 1
then the manifold is just G2. And if g[AoB] = g[A]g[B] for some X,Y-product AoB,
and h is an element of G2, then

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

so G2g is the manifold in question (clearly not a subgroup in general). Therefore,
if g and f are unravelled by the same X,Y-product, then gf-1 is in G2.

Some notes added 5 May 1997:
These ideas give us an explicit extension of G2 to
SO(8) --> G2 x S7 x S7 / Z2.
The 14-dimensional space of unit pairs (X,Y), modulo sign change, together with
the 14-dimensional G2, gives us the 28-dimensional SO(8).

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*B) = AXY*B = AWB,

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 S3. From the combination we can manufacture the 6-dimensional SO(4).

In the complex case the automorphism group is the 0-dimensional discrete group Z2, and
the W-manifold is S2, from the combination of which we can manufacture the 2-dimensional SO(2).

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.

Note of 8 May 1997:
This may finish this section looking at the relationship of SO(8) to the X,Y-product.
First two useful identities:

g0[AB] = Tog2To[A]Tog1To[B],
Tog0To[AB] = g1[A]g2[B].

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

g0[(AX)(Y*B)] = Tog2To[AX] Tog1To[Y*B] = (g0[A]g1[X]) (g2[Y*]g0[B]).

This is a fairly useful result. If g1[X] = ± X, and g2[Y*] = ± Y*,
then g0 is an X,Y-product automorphism.
And if g1[X] = ± 1, and g2[Y*] = ± 1,
then g0 unravels the X,Y-product.

For example, if X = U3 and Y* = U-1V3U, then a good guess for a g1 is
VLm VRn ULp URq,
so
g2 = VL-n VRm-n UL-q URp-q,
and
g0 = VLn-m VR-m ULq-p UR-p.
If we set g2[Y*] = 1, then need q = -1, p-q = -1, m-2n = -3, so p = -2. This gives us g1[X] = 1 = Vm+n, so m+n = 0, implying n = 1 and m = -1. Therefore,

g0 = VL2 VR UL UR2,

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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