  Let E be an integral lattice. We first discuss some general properties of an
SDC lattice, i.e., a sum of two diagonal copies of E in E \bot E. In
particular, we show that its group of isometries contains a wreath product. We
then specialize this study to the case of E = E_8 and provide a new and fairly
natural model for those rootless lattices which are sums of a pair of
EE_8-lattices. This family of lattices was classified in [7]. We prove that
this set of isometry types is in bijection with the set of conjugacy classes of
rootless elements in the isometry group O(E_8), i.e., those h \in O(E_8) such
that the sublattice (h - 1)E_8 contains no roots. Finally, our model gives new
embeddings of several of these lattices in the Leech lattice.
