4.4

We begin with the 78-dimensional exceptional Lie group .
As we mentioned in Section 3.4, there is a nice description of a
certain noncompact real form of as the group of collineations
of , or equivalently, the group of determinant-preserving
linear transformations of . But before going into these,
we consider the magic square constructions of the Lie algebra .
Vinberg's construction gives

Tits' construction, which is asymmetrical, gives

and also

The Barton-Sudbery construction gives

We can use any of these to determine the dimension of . For example, we have

Starting from the Barton-Sudbery construction and using the concrete
descriptions of and from equation
(3), we obtain

Using equation (4.2), we may rewrite this as

and it turns out that the summand is actually a Lie subalgebra of . This result can also be found in Adams' book [1], phrased as follows:

In fact, he describes the bracket in in terms of natural operations involving and its spinor representation . The funny-looking factor of comes from the fact that this spinor representation is complex. The bracket of an element of and an element of is another element of , defined using the obvious action of on this complex vector space.

If we define to be the simply connected group with Lie algebra
, it follows from results of Adams that the subgroup generated by
the Lie subalgebra
is isomorphic to
. This lets us define the **bioctonionic
projective plane** by

and conclude that the tangent space at any point of this manifold is isomorphic to .

Since is compact, we can put an -invariant Riemannian metric on the bioctonionic
projective plane by averaging any metric with respect to the action
of this group. It turns out [5] that the isometry group of this
metric is exactly , so we have

It follows that

Summarizing, we have 6 octonionic descriptions of :

where in each case the Lie bracket of is built from natural bilinear operations on the summands.

The smallest nontrivial representations of are 27-dimensional: in fact it has two inequivalent representations of this dimension, which are dual to one another. Now, the exceptional Jordan algebra is also 27-dimensional, and in 1950 this clue led Chevalley and Schafer [18] to give a nice description of as symmetries of this algebra. These symmetries do not preserve the product, but only the determinant.

More precisely, the group of determinant-preserving linear
transformations of turns out to be a noncompact real form of
. This real form is sometimes called , because its
Killing form has signature . To
see this, note that any automorphism of preserves the
determinant, so we get an inclusion

This means that is a compact subgroup of . In fact it is a maximal compact subgroup, since if there were a larger one, we could average a Riemannian metric group on with respect to this group and get a metric with an isometry group larger than , but no such metric exists. It follows that the Killing form on the Lie algebra is negative definite on its 52-dimensional maximal compact Lie algebra, and positive definite on the complementary 26-dimensional subspace, giving a signature of .

We saw in Section 3.4 that the projective plane structure of
can be constructed starting only with the determinant function
on the vector space . It follows that
acts as **collineations** on , that is, line-preserving
transformations. In fact, the group of collineations of is
precisely :

Moreover, just as the group of isometries of fixing a specific point is a copy of , the group of collineations fixing a specific point is . This fact follows with some work starting from equation (3.4), and it gives us a commutative square of inclusions:

where the groups on the top are maximal compact subgroups of those on the bottom. Thus in a very real sense, is to 9-dimensional Euclidean geometry as is to 10-dimensional Lorentzian geometry.

© 2001 John Baez