The octonions are fascinating in themselves, but the magic really starts
when we use them to construct the exceptional Jordan algebra
and its associated projective space, the octonionic projective plane.
The symmetry groups of these structures turn out to be exceptional Lie
groups, and triality gains an eerie pervasive influence over the
proceedings, since an element of consists of 3 octonions and
3 real numbers. Using the relation between normed division algebras and
trialities, we get an isomorphism
However, this picture of in terms of 8-dimensional Euclidean
geometry is just part of a bigger picture — a picture set in
10-dimensional Minkowski spacetime! If we regard as sitting
in the lower right-hand corner of , we get an isomorphism
This picture gives a representation of as linear
transformations of . Unfortunately, most of these
transformations do not preserve the Jordan product on . As we
shall see, they only preserve a lesser structure on : the determinant. However, the transformations coming from the subgroup
do preserve the Jordan product. We can
see this as follows. As a representation of ,
splits into 'space' and 'time':
This does not exhaust all the symmetries of , since there are other automorphisms coming from the permutation group on 3 letters, which acts on and in an obvious way. Also, any matrix acts by conjugation as an automorphism of ; since the entries of are real, there is no problem with nonassociativity here. The group is 36-dimensional, but the full automorphism group is much bigger: it is 52-dimensional. As we explain in Section 4.2, it goes by the name of .
However, we can already do something interesting with the automorphisms we have: we can use them to diagonalize any element of . To see this, first note that the rotation group, and thus , acts transitively on the unit sphere in . This means we can use an automorphism in our subgroup to bring any element of to the form
To understand , we need to understand projections in .
Here is where our ability to diagonalize matrices in via
automorphisms comes in handy. Up to automorphism, every projection in
looks like one of these four:
It follows that the points of the octonionic projective plane are
projections with trace 1 in , while the lines are projections
with trace 2. A calculation  shows that any projection
with trace 1 has the form
We use to stand for the set of points in the octonionic projective plane. Given any nonzero element with , we can normalize it and then use equation (3.4) to obtain a point . Copying the strategy that worked for , we can make into a smooth manifold by covering it with three coordinate charts:
We thus obtain the following picture of the octonionic projective plane. As a manifold, is 16-dimensional. The lines in are copies of , and thus 8-spheres. For any two distinct points in , there is a unique line on which they both lie. For any two distinct lines, there is a unique point lying on both of them. There is a 'duality' transformation that maps points to lines and vice versa while preserving this incidence relation. In particular, since the space of all points lying on any given line is a copy of , so is the space of all lines containing a given point!
To dig more deeply into the geometry of one needs another
important structure on the exceptional Jordan algebra: the determinant.
We saw in Section 3.3 that despite noncommutativity and
nonassociativity, the determinant of a matrix in is a
well-defined and useful concept. The same holds for ! We
can define the determinant of a matrix in by
The determinant is a cubic form on , so there is a unique symmetric
We have already seen that points of correspond to trace-1 projections in . Freudenthal  noticed that these are the same as elements with and . Even better, we can drop the equation as long as we promise to work with equivalence classes of nonzero elements satisfying , where two such elements are equivalent when one is a nonzero real multiple of the other. Each such equivalence class corresponds to a unique point of , and we get all the points this way.
Given two points and , their cross product is well-defined up to a nonzero real multiple. This suggests that we define a 'line' to be an equivalence class of elements , where again two such elements are deemed equivalent if one is a nonzero real multiple of the other. Freudenthal showed that we get a projective plane isomorphic to if we take these as our definitions of points and lines and decree that the point lies on the line if and only if . Note that this equation makes sense even though and are only well-defined up to nonzero real multiples.
One consequence of all this is that one can recover the structure of
as a projective plane starting from just the determinant on
: we did not need the Jordan algebra structure! However, to
get a 'duality' map switching points and lines while preserving the
incidence relation, we need a bit more: we need the nondegenerate
We thus obtain a very pretty setup for working with . If we
use the isomorphism between and its dual to reinterpret
the cross product as a map
For more on octonionic geometry, the reader is urged to consult the original papers of Freudenthal [35,36,37,38], as well as those of Jacques Tits [87,88] and Tonny Springer [80,81,82]. Unfortunately, we must now bid goodbye to this subject and begin our trip through the exceptional groups. However, we shall return to study the symmetries of and the exceptional Jordan algebra in Sections 4.2 and 4.4.
© 2001 John Baez