The E8 and D16+ models are positive even unimodular lattices

For $n=8$ or $16$, the homomorphism $ϵ:{\mathbb{Z}}^n\to \mathbb{Z}\mathrm{/}2\mathbb{Z}$, $ϵ(z)={\sum }_i{z}_i(\mathrm{m}\mathrm{o}\mathrm{d}2)$, is onto (it sends $e_1$ to $1$), and its kernel is $D_n$; hence $[{\mathbb{Z}}^n:D_n]=2$. The vectors $e_i-e_n$ for $1\le i<n$ and $e_{n-1}+e_n$ lie in $D_n$ and are linearly independent, so $D_n$ has rank $n$. By [F1], it has a $\mathbb{Z}$-basis, and [F2] gives its index as the absolute determinant of the basis matrix; [F3] identifies this with covolume, so $\mathrm{covol}(D_n)=2\mathrm{covol}({\mathbb{Z}}^n)=2$.

Since $2h_n=(1,\dots ,1)\in D_n$ for even $n$, ${\mathrm{\Lambda }}_n$ is the subgroup consisting of the two distinct cosets $D_n$ and $D_n+h_n$. It lies in $(1\mathrm{/}2){\mathbb{Z}}^n$, and $2{\mathrm{\Lambda }}_n$ is a subgroup of ${\mathbb{Z}}^n$ containing $2D_n$, so it has rank $n$. By [F1], choose a basis of $2{\mathrm{\Lambda }}_n$ as a subgroup of ${\mathbb{Z}}^n$; halving that basis gives a basis of ${\mathrm{\Lambda }}_n$, proving it is full. Its index over $D_n$ is $2$. Writing a basis of $D_n$ in a basis of ${\mathrm{\Lambda }}_n$ and applying [F2] gives the absolute determinant of that inclusion matrix as $2$; [F3] makes this the covolume ratio, so $\mathrm{covol}(D_n)=2\mathrm{covol}({\mathrm{\Lambda }}_n)$ and $\mathrm{covol}({\mathrm{\Lambda }}_n)=1$.

If $d\in D_n$, then $\mathrm{\parallel }d{\mathrm{\parallel }}^2={\sum }_i{d}_i^2\equiv {\sum }_i{d}_i\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}2)$, since $d_i^2\equiv d_i(\mathrm{m}\mathrm{o}\mathrm{d}2)$. For $d+h_n$, $\mathrm{\parallel }d+h_n{\mathrm{\parallel }}^2={\sum }_i{d}_i^2+{\sum }_i{d}_i+n\mathrm{/}4$; the first two sums have even total by the same congruence, and $n\mathrm{/}4$ is $2$ or $4$. Thus every vector in ${\mathrm{\Lambda }}_n$ has even integer norm. Polarization gives $2⟨x,y⟩=\mathrm{\parallel }x+y{\mathrm{\parallel }}^2-\mathrm{\parallel }x{\mathrm{\parallel }}^2-\mathrm{\parallel }y{\mathrm{\parallel }}^2\in 2\mathbb{Z}$ for $x,y\in {\mathrm{\Lambda }}_n$, so ${\mathrm{\Lambda }}_n$ is integral.

Integrality implies ${\mathrm{\Lambda }}_n\subseteq {\mathrm{\Lambda }}_n^{\ast }$. By [F3] and step 2.1, both full lattices have covolume $1$; applying [F2] in a basis of ${\mathrm{\Lambda }}_n^{\ast }$ shows their inclusion index is $1$, hence ${\mathrm{\Lambda }}_n={\mathrm{\Lambda }}_n^{\ast }$. This proves that $E_8={\mathrm{\Lambda }}_8$ and $D_{16}^{+}={\mathrm{\Lambda }}_{16}$ are positive even unimodular lattices.

The product inner product on ${\mathbb{R}}^8\oplus {\mathbb{R}}^8$ is positive definite, and the direct sum of two full lattices is full. A vector $(x,y)\in E_8\oplus E_8$ has norm $\mathrm{\parallel }x{\mathrm{\parallel }}^2+\mathrm{\parallel }y{\mathrm{\parallel }}^2\in 2\mathbb{Z}$, so the sum is even and integral. Its dual is $E_8^{\ast }\oplus E_8^{\ast }=E_8\oplus E_8$, since testing against $(u,0)$ and $(0,v)$ gives the two coordinate dual conditions. Thus the direct sum is positive even unimodular of rank $16$.