With 248 dimensions, is the biggest of the exceptional Lie
groups, and in some ways the most mysterious. The easiest way to
understand a group is to realize it as as symmetries of a structure one
already understands. Of all the simple Lie groups, is the only
one whose smallest nontrivial representation is the adjoint
representation. This means that in the context of linear algebra,
is most simply described as the group of symmetries of its own
Lie algebra! One way out of this vicious circle would be to describe
as isometries of a Riemannian manifold. As already mentioned,
is the isometry group of a 128-dimensional manifold called
. But alas, nobody seems to know how to define
without first defining . Thus this group remains
a bit enigmatic.
At present, to get our hands on we must start with its Lie
algebra. We can define this using any of the three equivalent magic
square constructions explained in the previous section. Vinberg's
Tits' construction gives
The Barton-Sudbery construction gives
We can use any of these to count the dimension of ; for example,
the last one gives
To emphasize the importance of triality, we can rewrite the
the Barton-Sudbery description of as
Here the Lie bracket is built from natural maps relating
and its three 8-dimensional irreducible representations. In particular,
is a Lie subalgebra, and the first copy of
acts on the first factor in
, while the second copy acts on the
second factor in each of these.
This has a pleasant resemblance to
the triality description of that we found in Section 4.2:
Now let us turn from 8-dimensional geometry to 16-dimensional geometry.
On the one hand, we have
On the other hand, we have seen that
Comparing these, it is natural to hope that
as a Lie subalgebra. In fact this is true! Even better, if we
restrict the right-handed spinor representation of to
, it decomposes as
so we obtain
or in more octonionic language,
where we use to mean the Lie algebra of skew-adjoint real-linear
transformations of the real inner product space .
The really remarkable thing about the isomorphism
is that the Lie bracket in is entirely built from natural maps involving
The first of these is the Lie bracket in , the second is the
action of on its right-handed spinor representation, and the
third is obtained from the second by duality, using the natural inner
product on and to identify these spaces with their
duals. In fact, this is a very efficient way to define .
If we take this approach, we must verify the Jacobi identity:
When all three of lie in this is just the Jacobi
identity for . When two of them lie in , it boils
down to fact that spinors indeed form a representation of .
Thanks to duality, the same is true when just one lies in . It
thus suffices to consider the case when all lie in .
This is the only case that uses anything special about the number 16.
Unfortunately, at this point a brute-force calculation seems to be
required. For two approaches that minimize the pain, see the books by
Adams  and by Green, Schwarz and Witten . It
would be nice to find a more conceptual approach.
Starting from , we can define to be the simply-connected
Lie group with this Lie algebra. As shown by Adams , the
subgroup of generated by the Lie subalgebra
. This lets us define the octooctonionic
projective plane by
The tangent space at any point
of this manifold is isomorphic to
This partially justifies calling it 'octooctonionic projective plane',
though it seems not to satisfy the usual axioms for a projective plane.
We can put an -invariant Riemannian metric on the octooctonionic
projective plane by the technique of averaging over the group action.
It then turns out  that
Summarizing, we have the following octonionic descriptions of