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## 4.5 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 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 : 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 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  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  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.

Next: E8 Up: Exceptional Lie Algebras Previous: E6