  Starting from basic identities of the group E8, we perform progressive
reductions, namely decompositions with respect to the maximal and symmetric
embeddings of E7xSU(2) and then of E6xU(1). This procedure provides a
systematic approach to the basic identities involving invariant primitive
tensor structures of various irreprs. of finite-dimensional exceptional Lie
groups. We derive novel identities for E7 and E6, highlighting the E8 origin of
some well known ones. In order to elucidate the connections of this formalism
to four-dimensional Maxwell-Einstein supergravity theories based on symmetric
scalar manifolds (and related to irreducible Euclidean Jordan algebras, the
unique exception being the triality-symmetric N = 2 stu model), we then derive
a fundamental identity involving the unique rank-4 symmetric invariant tensor
of the 0-brane charge symplectic irrepr. of U-duality groups, with potential
applications in the quantization of the charge orbits of supergravity theories,
as well as in the study of multi-center black hole solutions therein.

High Energy Physics - Theory