# Involution of $E_{8}$ lattice

Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are isomorphic). I am sure that this has been studied in some literature. I would appreciate it if anyone could let me know the classification or some reference.

The isometry group of the $E_8$ lattice is the same as the $E_8$ Weyl group (a fact that doesn't always hold for other weight lattices). So you're asking for conjugacy classes of order two elements in the $E_8$ Weyl group. Have you tried looking into the atlas of simple groups? –  André Henriques Aug 11 '12 at 22:30
Not necessarily isomorphic; not even necessarily the same rank, e.g. reflection about a root has $L^+ \cong E_7$ and $L^- \cong A_1$ (and even more trivially central reflection has $L^+ = \lbrace 0 \rbrace$ and $L^- = E_8$). But it is possible for both $L^+$ and $L^-$ to be isomorphic with $D_4$. –  Noam D. Elkies Aug 12 '12 at 2:34