A typical $E_8$ lattice is of the form of $E_8$ Cartan matrix:
−1 2 −1 0 0 0 0 0
0 −1 2 −1 0 0 0 0
0 0 −1 2 −1 0 0 0
0 0 0 −1 2 −1 0 −1
0 0 0 0 −1 2 −1 0
0 0 0 0 0 − 1 2 0
0 0 0 0 −1 0 0 2
Which $E_8$ Cartan matrix has all integer entries and with determinant $=1$.
I find an expression of typical Leech lattice with all integer entries, https://en.wikipedia.org/wiki/Leech_lattice#Using_the_binary_Golay_code
$$ 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0\\ 2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0\\ 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0\\ 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0\\ 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0 0\\ 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0\\ 2 0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0\\ 2 0 0 2 2 2 0 0 2 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0\\ 2 2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0 0\\ 0 2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0\\ 0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0\\ 0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0\\ -3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 $$ But this matrix has determinant $68719476736=2^{36}$.
In order to have the rank-24 matrix has determinant $1$, we need to multiply it by $$\frac{1}{\sqrt{8}}.$$
- My question is that:
Why the $E_8$ matrix can have simultaneous integer entries with unit determinant. But that the Leech matrix cannot have simultaneous integer entries with unit determinant or integer determinant? regardless whatever overall scalings - what this negative statement imply for Leech matrix, in contrast with the $E_8$ matrix?
Let me call $L$ the Leech lattice and $A$ the matrix you've given. $\frac1{\sqrt8}A$ is the matrix that gives you an embedding of $L$ into $\mathbb{R}^{24}$ (with the standard scalar product).
Therefore, the Gram matrix associated to this embedding is $\frac18AA^t$, which is: $$ \tiny \left( \begin{array}{cccccccccccccccccccccccc} 8 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 4 & 4 & 4 & 2 & 4 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 0 & 0 & 0 & -3\\ 4 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 1 & 2 & 1 & 0 & 0 & -1\\ 4 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 1 & 1 & 0 & 0 & -1\\ 4 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 1 & 1 & 0 & 0 & -1\\ 4 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 0 & 0 & -1\\ 4 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 1 & 0 & 0 & 0 & -1\\ 4 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 1 & 2 & 0 & 0 & 0 & -1\\ 2 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 0 & 0 & 1\\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 4 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & -1\\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 0 & 1 & 0 & -1\\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1 & 2 & 1 & 2 & 1 & 0 & 0 & 1 & -1\\ 2 & 2 & 2 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 4 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1\\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 4 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & -1\\ 2 & 2 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 1 & 1\\ 2 & 1 & 2 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 2 & 1\\ 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 4 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1\\ 4 & 2 & 2 & 2 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 4 & 2 & 2 & 2 & 1 & 1 & 1 & -1\\ 2 & 1 & 2 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 2 & 2 & 1 & 1\\ 2 & 1 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 2 & 1 & 2 & 1\\ 2 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 4 & 2 & 1 & 1 & 1\\ 0 & 1 & 1 & 1 & 1 & 0 & 0 & 2 & 1 & 0 & 0 & 2 & 1 & 2 & 2 & 2 & 1 & 2 & 2 & 2 & 4 & 2 & 2 & 2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 1 & 2 & 4 & 2 & 2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 1 & 2 & 2 & 4 & 2\\ -3 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & -1 & -1 & -1 & 1 & -1 & 1 & 1 & 1 &-1 & 1 & 1 & 1 & 2 & 2 & 2 & 4 \end{array} \right). $$