Diagonal lattices and rootless EE8 pairs

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⊥E. In particular, we show that its group of isometries contains a wreath product. We then specialize this study to the case of E=E8 and provide a new and fairly natural model for those rootless lattices which are sums of a pair of EE8-lattices. This family of lattices was classified in Griess Jr. and Lam (2011) [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(E8), i.e., those h∈O(E8) such that the sublattice (h−1)E8 contains no roots. Finally, our model gives new embeddings of several of these lattices in the Leech lattice.