TITLE: 24 vectors in Leech lattice having scalar product $\frac{1}{4}$ pairwise
QUESTION [10 upvotes]: Two vectors from Leech lattice - as defined on wikipedia - have scalar product $\pm 32,\pm 16, \pm 8$ or $0$. Do there exist 24 vectors having scalar product 8 pairwise ? When we consider unit vectors in Leech lattice then scalar product is $\frac{1}{4}$ pairwise.
Motivation for this question is the fact that for 2b involution there are 196560 involutions in Monster group 2a conjugacy class which generate extraspecial group $2^{1+24}$. Modulo its center elements in this group correspond to Leech vectors. I believe vectors having scalar product 8 correspond to not commuting elements.
What I was able to achieve is that for $\Lambda_{16}$ sublattice there are at most 9 such vectors. The arguments are following. This sublattice can be seen as union of 18 disjoint $E_8$ lattices (see Wilson definition with octonions which refer to earlier Dixon definition). Given two vectors having scalar product 8 ($\frac {1}{2}$ in Wilson version where Leech vector is of legth $\sqrt 2$) must lie in two different E8s.
For Baby monster 2b element there are 4600 elements in 2a commuting with it. In Leech lattice language for fixed vector there are 4600 vectors having scalar product 16 with it.
Regards,

REPLY [2 votes]: Here are 24 vectors in classic Leech lattice having pairwise scalar product 8. I obtained it from $(-5, 1^{23})$ using this.
$\begin{bmatrix}
-1& 1& 1& 1& 1&-1& 1&-1& 1& 1&-1&-1& 1& 1&-1&-1& 1&-1& 1&-1&-1&-1&-1&-3 \\
\dots\\
 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1&-3 
\end{bmatrix}$
The dots $\dots$ should be replaced by 22 vectors obtained from the first one by permutation $(1,2,...,23)$.
As a consequence we can obtain following useful 
Fact
For each element $p$ of order $23$ in $M_{23}$ there exist dodecad $d$ such that dodecads $d, pd, p^2d,...,p^{22}d$ intersect in six points pairwise.