E8

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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 construction gives



To emphasize the importance of triality, we can rewrite the the Barton-Sudbery description of as

Now let us turn from 8-dimensional geometry to 16-dimensional geometry. On the one hand, we have

The really remarkable thing about the isomorphism



Starting from , we can define to be the simply-connected Lie group with this Lie algebra. As shown by Adams [1], the subgroup of generated by the Lie subalgebra is . This lets us define the octooctonionic projective plane by



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 [5] that



Summarizing, we have the following octonionic descriptions of :

is given by

Theorem 8. The compact real form ofis given by where in each case the Lie bracket on is built from natural bilinear operations on the summands. where in each case the Lie bracket onis built from natural bilinear operations on the summands.

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© 2001 John Baez