Steinitz classes of unimodular lattices

For each prime l with l≡7(mod 8), we define an action of the ring O=Z[12(1+−l)] on the unimodular lattice Dl+1+ using a Paley matrix. We determine the isomorphism class of Dl+1+ as an O-module. In particular we show that unless l=7, Dl+1+ is not a free O-module. We note a consequence for the Leech lattice.