A description of the isometry group $O(U\oplus E_8)$?

Question

Are there any good description of the isometry group $O(U\oplus E_8)$? Here $U$ denotes the hyperbolic lattice and $E_8$ the root lattice of type $E_8$.

Answer 1

Yes. See Lattices and Codes by Wolgang Ebeling. In the second edition, this is Exercise 4.4 on page 134. I do not believe this information was in the first edition; further, there is a third edition now. Anyway, the isometry group, or automorphism group, is generated by reflections in the roots of $L$ and $\pm 1.$ Essentially you use the proof of Theorem 4.6, due to Conway. The same techniques shows that, for any integral even lattice with covering radius strictly below $\sqrt 2,$ the class number of the lattice is one. In particular, it is not necessary to have unimodularity for this latter result.