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 --> V_L²V_RU_LU_R²[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^~ = U_R^-2U_L^-1V_R^-1V_L^-2 G₂ V_L²V_RU_LU_R².

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{V^k(U^pAU^q)V^j}W^m;
X = U^p(V^jW^2m-nV^j-k)U^q-p;
Y^* = U^p-q(V^k-jW^2n-mV^k)U^p.

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] = T_og₂T_o[A]T_og₁T_o[B],
T_og₀T_o[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)] = T_og₂T_o[AX] T_og₁T_o[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
V_L^m V_Rⁿ U_L^p U_R^q,
so
g₂ = V_L^-n V_R^m-n U_L^-q U_R^p-q,
and
g₀ = V_L^n-m V_R^-m U_L^q-p U_R^-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 = V^m+n, so m+n = 0, implying n = 1 and m = -1. Therefore,

g₀ = V_L² V_R U_L U_R²,

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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