E8 ⊕ E8 and D16+ rank-16 lattice models

For any positive integer $n$, use the following coordinate labels: for a vector $x:\{0,\dots ,n-1\}\to \mathbb{R}$, write $x=(x_1,\dots ,x_n)$ with $x_i:=x(i-1)$ for $1\le i\le n$. Thus ${\mathbb{Z}}^n$ consists of those vectors whose labeled coordinates are integers, and the standard inner product is $⟨x,y⟩={\sum }_{i=1}^n{x}_i{y}_i={\sum }_{k=0}^{n-1}x(k)y(k)$. Write $e_i$ for the vector whose $i$-th labeled coordinate is $1$ and whose other coordinates are $0$.

For $n\in \{8,16\}$ let $D_n:=\{z=(z_1,\dots ,z_n)\in {\mathbb{Z}}^n:{\sum }_{i=1}^n{z}_i\in 2\mathbb{Z}\}$ and $h_n:=(1\mathrm{/}2,\dots ,1\mathrm{/}2)\in {\mathbb{R}}^n$. In ${\mathbb{R}}^8$ with its standard inner product define $E_8:=D_8\cup (D_8+h_8)$. In ${\mathbb{R}}^{16}$ with its standard inner product define $D_{16}^{+}:=D_{16}\cup (D_{16}+h_{16})$. Define $E_8\oplus E_8:=\{(x,y)\in {\mathbb{R}}^8\oplus {\mathbb{R}}^8:x\in E_8,y\in E_8\}$ with the orthogonal product inner product. These are specified full lattices; their evenness and unimodularity, and the nonisometry of the two rank-16 models, are separate mathematical claims.

Positive-definite even unimodular lattice

The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$