A question on an involution of $E_8$ lattice There exits an involution $\iota$ of the $E_8$ lattice such that $(E_8)^{\pm} \cong D_4$, where $(E_8)^{\pm}$ denotes the $\pm$ eigen-lattice of the involution $\iota$. Could someone kindly give me an explicit description of such involution $\iota$? 
Thank you very much. 
 A: If you construct $E_8$ as 
$\lbrace{(x_1,\ldots,x_8) \in D_8^*: \sum_{i=1}^8 x_i \in 2{\bf Z} \rbrace}$
then you can use an involution such as 
 $(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8) \leftrightarrow
 (x_1,x_2,x_3,x_4, -x_5,-x_6,-x_7,-x_8)$.
Recall that $D_8^* = {\bf Z}^8 \cup ({\bf Z}+\frac12)^8$;
the plus and minus eigenspaces of our involution are
$\lbrace(x_1,x_2,x_3,x_4,0,0,0,0)\rbrace$ and
$\lbrace(0,0,0,0,x_5,x_6,x_7,x_8)\rbrace$,
and those contain no $({\bf Z}+\frac12)^8$ vectors,
leaving only a copy of $D_4$ in each eigenspace.
A: If you get $E_8$ as the "Construction A" lattice associated to the
extended Hamming [8,4,4] code, then you can choose a weight-$4$ codeword $c$,
and use the involution that multiplies by $-1$ each of the coordinates in the
support of $c$.  Then the plus and minus eigenlattices are each identified
with the "Construction A" lattice associated to the $[4,1,4]$ repetition code,
which is $D_4$.  [If $C$ is a binary linear code of length then
$$
\lbrace 2^{-1/2} v : v \in {\bf Z}^n,  v \bmod 2 \in C \rbrace
$$
is the associated lattice; see the section on Construction A in
Conway and Sloane's Sphere Packings, Lattices and Groups.
If you instead invert four coordinates that are not the support of a codeword
then each eigenlattice is $A_1^4$.]
A: Depends on how you're constructing $E_8$.  One approach is to start from
$D_4 \oplus D_4$ and reconstruct $E_8$ as the sublattice of 
$D_4^* \oplus D_4^*$ consisting of pairs of vectors that are
congruent ${\rm mod}\phantom.D_4$.  Check that this is an even unimodular lattice,
and thus isomorphic to $E_8$.  The involution is $(v,w) \leftrightarrow (v,-w)$.
