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## 4.5

Next we turn to the 133-dimensional exceptional Lie group . In 1954, Freudenthal [37] 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

and also

The Barton-Sudbery version gives

Starting from equation (4) and using the fact that is 3-dimensional, we obtain the elegant formula

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

Using equation (4.2), we may rewrite this as

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 [1]:

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 [1] that the subgroup generated by the Lie subalgebra is isomorphic to . This lets us define the quateroctonionic projective plane by

and conclude that the tangent space at any point of this manifold is isomorphic to . We can put an -invariant Riemannian metric on this manifold by the technique of averaging over the group action. It then turns out [5] that

and thus

Summarizing, we have the following 7 octonionic descriptions of :

Theorem 7   The compact real form of is given by

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

Before the magic square was developed, Freudenthal [37] 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 [44] has extended this to a relation between 10-dimensional conformal geometry and , and in work with Kilian Koepsell and Hermann Nikolai [45] has explicated how this is connected to Freudenthal triple systems.

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