Next we turn to the 133-dimensional exceptional Lie group .
In 1954, Freudenthal  described this group as
the automorphism group of a 56-dimensional octonionic structure now
called a 'Freudenthal triple system'. We sketch this idea
below, but first we give some magic square constructions.
Vinberg's version of the magic square gives
Tits' version gives
The Barton-Sudbery version gives
Starting from equation (4) and using the fact that
is 3-dimensional, we obtain the elegant
This gives an illuminating way to compute the dimension of :
Starting from equation (5) and using the concrete
descriptions of and from equation
(3), we obtain
Though not obvious from what we have done, the direct summand
here is really a Lie subalgebra of . In
less octonionic language, this result can also be found in Adams' book
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
representation is quaternionic. The bracket of an element of and
an element of is the element of defined using
the natural action of on this space.
If we let be the simply connected group with Lie
algebra , it follows from results of Adams  that the
subgroup generated by the Lie subalgebra
This lets us define the quateroctonionic projective plane by
and conclude that the tangent space at any point of this manifold is
. We can put
an -invariant Riemannian metric on this manifold by the technique
of averaging over the group action. It then turns out  that
Summarizing, we have the following 7 octonionic descriptions of :
where in each case the Lie bracket of is built from
natural bilinear operations on the summands.
Before the magic square was developed, Freudenthal 
used another octonionic construction to study . The smallest
nontrivial representation of this group is 56-dimensional. Freudenthal
realized we can define a 56-dimensional space
and equip this space with a symplectic structure
and trilinear product
such that the group of linear transformations preserving both these
structures is a certain noncompact real form of , namely
. The symplectic structure and trilinear product on
satisfy some relations, and algebraists have made these into the
definition of a 'Freudenthal triple system' [10,32,67].
The geometrical significance of this rather complicated sort of structure
has recently been clarified by some physicists working on string theory.
At the end of the previous section, we
mentioned a relation between 9-dimensional Euclidean geometry and
, and a corresponding relation between 10-dimensional Lorentzian
geometry and . Murat Günaydin  has
extended this to a relation between 10-dimensional conformal
geometry and , and in work with Kilian Koepsell
and Hermann Nikolai  has explicated how this is connected
to Freudenthal triple systems.