Two lattices and in the same vector space are called neighbours if their intersection is of index two in both and .
In 1957, Martin Kneser gave a method to find all unimodular lattices (of the same dimension and signature) starting from one such unimodular lattice, finding all its neighbours, and repeating this with the new lattices obtained.
In other words, Kneser’s neighbourhood graph, with vertices the unimodular lattices (of fixed dimension and signature) and edges between them whenever the lattices are neighbours, is connected.

Martin Kneser (1928-2004) – Photo Credit
Last time, we’ve constructed the Niemeier lattice from the binary Golay code
With hindsight, we know that is the unique neighbour of the Leech lattice in the Kneser neighbourhood graph of the positive definite, even unimodular -dimensional lattices, aka the Niemeier lattices.
Let’s try to construct the Leech lattice from by Kneser’s neighbour-finding trick.

Sublattices of of index two are in one-to-one correspondence with non-zero elements in . Take and such that the inner product is odd, then
is an index two sublattice because . By definition is even for all and therefore . We have this situation
and , with the non-zero elements represented by . That is,
This gives us three lattices
and all three of them are unimodular because
and is of index in .
Now, let’s assume the norm of , that is, . Then, either the norm of is odd (but then the norm of must be even), or the norm of is even, in which case the norm of is odd.
That is, either or is an even unimodular lattice, the other one being an odd unimodular lattice.
Let’s take for and the vectors and , then
Because is odd, we have that
is an even unimodular lattice, which is the Leech lattice, and
is an odd unimodular lattice, called the odd Leech lattice.

John Leech (1926-1992) – Photo Credit
Let’s check that these are indeed the Leech lattices, meaning that they do not contain roots (vectors of norm two).
The only roots in are the roots of and they are of the form , but none of them lies in as their inproduct with is one. So, all non-zero vectors in have norm .
As for the other part of and
From the description of it follows that every coordinate of a vector in is of the form
with , with the second case instances forming a codeword in . In either case, the square of each of the coordinates is , so the norm of such a vector must be , showing that there are no roots in this region either.
If one takes for a vector of the form where or and , takes as before, and repeats the construction, one gets the other Niemeier-neighbours of , that is, the lattices , and .
For one needs a slightly different argument, see section 0.2 of Richard Borcherds’ Ph.D. thesis.