Lattices like the Leech lattice (Q913845)

The main result of this paper (which applies to the Coxeter-Todd lattice of dimension 12 and the Barnes-Wall lattice of dimension 16) says the following. Let G be a group of automorphisms of the Leech lattice \(\Lambda\) such that the sublattice \(\Lambda '\) of vectors of \(\Lambda\) fixed by G has no roots, and let R be the Lorentzian lattice that is the sum of \(\Lambda '\) and the unimodular hyperbolic plane. Then R has a norm 0 vector c such that the simple roots of the reflection group of R are exactly the roots r of R such that the inner product (r,c) is negative and divides (r,v) for all vectors v of R. The group Aut(R) is isomorphic to a split extension of its reflection group by the group of affine automorphisms of \(\Lambda '\).