A quaternionic approach to E 7

The classical Lie algebras over the complex numbers are all derived from associative algebras by defining a Lie bracket [A,B] = AB−BA, but the five exceptional Lie algebras are often defined directly as Lie algebras. In four cases, however, an alternative approach is available to construct both the algebra and the corresponding Lie group. It is well-known that the Lie group G2 is naturally defined as the automorphism group of the octonions (Cayley numbers), and F4 as the automorphism group of the exceptional Jordan algebra, a 27-dimensional algebra of 3× 3 Hermitian matrices over octonions with product AB +BA. Moreover, E6 may be defined as the group of linear maps on the Jordan algebra which preserve a certain ‘determinant’ (but not the algebra product). This leaves just E7, which has a 56-dimensional representation whose structure is hard to describe. But recognising that this is really a 28dimensional quaternionic representation simplifies things significantly, and reveals tantalising glimpses of an underlying 7-dimensional structure over the tensor product of two quaternion algebras.