Development of a unified tensor calculus for the exceptional Lie algebras

The uniformity of the decomposition law, for a family F of Lie algebras which includes the exceptional Lie algebras, of the tensor powers ad of their adjoint representations ad is now well-known. This paper uses it to embark on the development of a unified tensor calculus for the exceptional Lie algebras. It deals explicitly with all the tensors that arise at the n = 2 stage, obtaining a large body of systematic information about their properties and identities satisfied by them. Some results at the n = 3 level are obtained, including a simple derivation of the the dimension and Casimir eigenvalue data for all the constituents of ad. This is vital input data for treating the set of all tensors that enter the picture at the n = 3 level, following a path already known to be viable for a1 ∈ F . The special way in which the Lie algebra d4 conforms to its place in the family F alongside the exceptional Lie algebras is described.