Consider two unimodular matrices $M_{E_8}$ and $M_{\operatorname{Spin}(32)/\mathbb{Z}_2}$:
$$M_{E_8} = \begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \cr -1 & 2 & -1 & 0 & 0 & 0 & -1 & 0 \cr 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \cr 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \cr 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \cr 0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 \cr 0 & -1 & 0 & 0 & 0 & 0 & 2 & -1 \cr 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 \cr \end{pmatrix}.$$
$$ M_{\operatorname{Spin}(32)/\mathbb{Z}_2} = \left( \begin{array}{cccccccccccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 4 \end{array} \right).$$ The $\operatorname{Spin}(32)/\mathbb{Z}_2$ means that a basis for the lattice is given by the roots of $SO(32)$, but with the root corresponding to the vector representation replaced by the weight of one of the spinor representations.
It seems that the following two unimodular matrices $M_{E_8} \oplus M_{E_8}$ and $M_{\operatorname{Spin}(32)/\mathbb{Z}_2}$ are related by certain $SL(16,\mathbb{Z})$ basis change. That is called this basis change matrix $U$.
question 1. If it is true, can we rewrite: $$M_{E_8} \oplus M_{E_8}=U M_{\operatorname{Spin}(32)/\mathbb{Z}_2} U^\dagger?$$ with $U \in SL(16,\mathbb{Z})$. (optional: what is this $U$?)
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question 2. Is it true that the canonical form of this unimodular indefinite symmetric integral matrices can be rewritten as $$\bigl( {\begin{smallmatrix} 1 &0 \\ 0 & -1 \end{smallmatrix}} \bigl) \oplus \bigl( {\begin{smallmatrix} 1 &0 \\ 0 & -1 \end{smallmatrix}} \bigl) \oplus \bigl( {\begin{smallmatrix} 1 &0 \\ 0 & -1 \end{smallmatrix}} \bigl) \oplus \cdots$$ and $$ \bigl( {\begin{smallmatrix} 0 &1 \\ 1 & 0 \end{smallmatrix}} \bigl) \oplus \bigl( {\begin{smallmatrix} 0 &1 \\ 1 & 0 \end{smallmatrix}} \bigl)\oplus \bigl( {\begin{smallmatrix} 0 &1 \\ 1 & 0 \end{smallmatrix}} \bigl) \oplus \cdots $$ and a set of all positive (or negative) coefficients $E_8$ lattices, $M_{E_8}$?