Page 10.

What follows is an expansion of the talk I gave at Corinne and Tevian's Octoshop II
meeting in Corvallis, Oregon in 1996. I've entitled it:

Unravelling Triality with the X,Y-Product

Background material on triality can be found in many places, including my book.
Briefly, an 8-dimensional real orthogonal space V has a Clifford algebra representable by the
algebra of real 16x16 matrices, which has dimension 16² = 2⁸. The
spinor space on which this acts is a 16-dimensional real space. The Lie algebra so(8)
appears in the Clifford algebra as the 28-dimensional collection of bivectors. It splits the
spinors in two, an 8-dimensional halfspinor S⁺, and an 8-dimensional halfspinor S^-.
So, including the initial space V we have 3 real 8-dimensional spaces, all because
((2⁸)^1/2)/2 = 8. SO(8) actions on any one of these 3 spaces induces actions
on the other 2, and these 3 representations are related by a cyclical map. This is an example
of triality, the full potential of which is only realized in dimension 8.

The octonions, O, and triality are part of the same mathematical resonance. I'm going
develop a connection here, but first some notational background. Let U be a unit octonion, and
A an arbitrary octonion. Define the maps

A --> U_L[A] = UA,
A --> U_R[A] = AU.

These are SO(8) maps on the real 8-dimensional space of octonions. Define

A --> T_o[A] = A^*

to be the anti-automorphic octonion conjugation map.

Ok, so here in a nutshell is what I plan to do here. I'm going to take some g_o
in SO(8), perform the triality map on it twice to get two more elements of SO(8),
g₁ and g₂. For any two octonions A and B these will satisfy

g_o[A]g₁[B] = T_og₂T_o[AB];
g₁[A]g₂[B] = T_og_oT_o[AB];
g₂[A]g_o[B] = T_og₁T_o[AB].

That is, SO(8) is almost automorphic on O, and would be if it didn't get all
ravelled up by triality. The subgroup of SO(8) for which g_o = g₁ = T_og₂T_o, etc., is
G₂, the actual automorphism group of O. Any g in G₂ satisfies g[A]g[B] = g[AB].

The X,Y-product can unravel SO(8). In particular, for any element g of SO(8) there
will be unit octonions X and Y such that if we set AoB = (AX)(Y^*B) (which is
an alternate definition of multiplication on O), then

g[A]g[B] = g[AoB].

If AoB = AB (when X = Y = ± 1), then g is in G₂, otherwise not, even though the map is now "unravelled"
in this equation, ie., does not require the triality map to achieve equality.

Ok, now to carry on to some explicit examples.

One of the Moufang identities implies

(UA)(BU) = U(AB)U.

This one can be rewritten as follows when U is a unit octonion:

(U_L[A])(U_R[B]) = T_oU_L^*U_R^*T_o[AB].

That is,

U_L --> U_R --> U_L^*U_R^*

is a triality triple. The action of triality on the last or these three maps takes us
back to the first.

Using the ideas developed above we arrive via triality at two more identities:

(AU)(U^*BU^*) = (AB)U^*;
(U^*AU^*)(UB) = U^*(AB).

As another example, the following maps are a triality triple:

U_LU_R^* --> U_LU_R² --> U_L^*2U_R^*.

This results in the three identities:

(UAU^*)(UBU²) = U(AB)U²;
(UAU²)(U^*2BU^*) = U(AB)U^*;
(U^*2AU^*)(UBU^*) = U^*2(AB)U^*.

Ok, let's do some triality unravelling. In the second of the previous three equations
replace A by AU^*3, and B by U³B. This yields:

(UAU^*)(UBU^*) = g[A]g[B] = U((AU^-3)(U³B))U^* = g[AoB],

where in this case the X,Y-product unravelling the map g has X = Y = U^-3 (so it is
an X-product). We can rewrite this identity as follows:

U^*((UAU^*)(UBU^*))U = (AU^-3)(U³B) = g^-1[g[A]g[B]].

This gives us a clear isomorphism from O to its variant with the product AoB, call it O^~:

A --> U^*AU.

Moreover it is now simple to prove that if G₂^~ is the automorphism group of O^~,
then it is related to our original copy of G₂ by:

G₂^~ = U_RU_L^*G₂U_LU_R^*.

And this will about finish it for this page. Bigger and more general results will follow on page 11.
Notice that if U³ = ±1, then the equations above again prove that sixth roots of unity
generate elements of G₂ of the form U_LU_R^*.

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