In following question on MathOverflow I received construction of new Leech lattice provided by Noam Elkies. Let's call it $(E)$. This Leech lattice has nice feature that there is easy to see $24$ vectors having pairwise scalar product $\frac{1}{4}$ (in case when we have unit vectors in Leech lattice). As defined in answer length of $(E)$ lattice vector is $\sqrt {48}$. The $24$ vectors having pairwise $\frac{1}{4}$ scalar product are $-5,1^{23}$ and permutations.

I would like to map this lattice to classical Leech lattice. Let's call it $(C)$. The definition of classical Leech lattice can be found for example here. The length of $(C)$ lattice vector is $\sqrt {32}$.

My question is how to map lattice $(E)$ to lattice $(C)$. I thought it is easy, but it isn't. We need to select proper basis in one lattice and in another. It should be done in a way, so given Leech lattice is oriented in the same way to one as to the second. This is not easy task. It really requires much understanding what are the symmetries of Leech lattice.

For example I observed that permutation $(1..23)$ of first 23 coordinates preserve both lattices. There is just one fixed vector by this automorphism. Next I tried to find somehow remaining $23$ vectors which would be oriented the same way but no luck so far.

I asked this question on Stack Exchange $12$ days ago without any answer. Therefore I am trying here now.

At this moment I am considering following idea.

- Find $E_8$ sublattice.
- Filter perpendicular $\Lambda_{16}$.
- Decompose $\Lambda_{16}$ to two perpendicular $E_8$. Now we have three perpendicular $E_8$ in Leech lattice $L$.
- Find orthogonal basis in each of three $E_8$ built from lattice vectors. We need to do it in good way so combined basis of $24$ vectors can be used for mapping between two Leech lattices.

Regards,