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:

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 16

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

So, including the initial space V we have 3 real 8-dimensional spaces, all because

((2

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,

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

A --> U

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

A --> T

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

in SO(8), perform the triality map on it twice to get two more elements of SO(8),

g

g

g

g

That is, SO(8) is almost automorphic on

ravelled up by triality. The subgroup of SO(8) for which g

G

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

an alternate definition of multiplication on

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

If AoB = AB (when X = Y = ± 1), then g is in G

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

That is,

U

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

(U

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

U

This results in the three identities:

(UAU

(UAU

(U

Ok, let's do some triality unravelling. In the second of the previous three equations

replace A by AU

(UAU

where in this case the X,Y-product unravelling the map g has X = Y = U

an X-product). We can rewrite this identity as follows:

U

This gives us a clear isomorphism from

A --> U

Moreover it is now simple to prove that if G

then it is related to our original copy of G

G

And this will about finish it for this page. Bigger and more general results will follow on page 11.

Notice that if U

generate elements of G

Page 9. | Page 11. | Contents | Octoshop III page.