There are several rank-8 square matrices known to be related to $E_8$:

1. Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix

$$M_1=\left [\begin{array}{rr}
 2 & -1 &  0 &  0 &  0 &  0 &  0 & 0 \\
-1 &  2 & -1&  0 &  0 &  0 &  0 & 0  \\
 0 & -1 &  2 & -1 &  0 &  0 &  0 & 0  \\
 0 &  0 & -1 &  2 & -1 &  0 &  0 & 0  \\
 0 &  0 &  0 & -1 &  2 & -1 &  0 & -1  \\
 0 &  0 &  0 &  0 & -1 &  2 & -1 & 0  \\
 0 &  0 &  0 &  0 &  0 & -1 &  2 & 0  \\
 0 &  0 & 0 &  0 &  -1 &  0 &  0 & 2
\end{array}\right ]$$

2. One choice of simple roots of $E_8$ is given by the rows of the following matrix
https://en.wikipedia.org/wiki/E8_(mathematics)#Simple_roots

$$M_2=\left [\begin{array}{rr}
1&-1&0&0&0&0&0&0 \\
0&1&-1&0&0&0&0&0 \\
0&0&1&-1&0&0&0&0 \\
0&0&0&1&-1&0&0&0 \\
0&0&0&0&1&-1&0&0 \\
0&0&0&0&0&1&1&0 \\
-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
0&0&0&0&0&1&-1&0 
\end{array}\right ]$$

3. $E_8$ lattice https://en.wikipedia.org/wiki/E8_lattice#Properties


$$M_3=\left[\begin{matrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 1/2 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 1/2 \\
0 & 0 & 1 & -1 & 0 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 1 & -1 & 0 & 0 & 1/2 \\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 1/2 \\
0 & 0 & 0 & 0 & 0 & 1 & -1 & 1/2 \\
0 & 0 & 0 & 0 & 0 & 0 & 1  & 1/2 \\
0 & 0 & 0 & 0 & 0 & 0 & 0  & 1/2
\end{matrix}\right]$$

My question is that 

- What are relations between different $E_8$ matrices? $M_1, M_2, M_3$ (Of course, they are related to  $E_8$.) Could they be transformed to one from another, vice versa? (like $GL(8,\mathbf{Z})$ What are the meanings of these transformations?