It's worth noting this fact. Suppose $L$, $L'$ are even unimodular Euclidean lattices, like $\mathrm{E}_8= \Gamma^8$ or $\Gamma^{16}$ or any direct sum of these, but also the Leech lattice and others in 24 dimensions. Let $\Gamma^{1,1}$ be the even unimodular Lorentzian lattice in 2 dimensions. Then

$$L \oplus \Gamma^{1,1} \cong L' \oplus \Gamma^{1,1}$$

The reason is that there's only one even unimodular Lorentzian lattice in dimensions $8n + 2$ (and none in other dimensions), up to isomorphism.

So, the fact that $\mathrm{E}^8 \oplus \mathrm{E}^8$ and $\Gamma^{16}$ become isomorphic when you take their direct sum with $\Gamma^{1,1}$ is an instance of this general pattern.

It's worth noting this fact. Suppose $L$, $L'$ are even unimodular Euclidean lattices, like $\mathrm{E}_8= \Gamma^8$ or $\Gamma^{16}$ or any direct sum of these, but also the Leech lattice and 23 others in 24 dimensions, and at least 80,000,000,000,000,000 others in 32 dimensions, etc. Let $\Gamma^{1,1}$ be the even unimodular Lorentzian lattice in 2 dimensions. Then

$$L \oplus \Gamma^{1,1} \cong L' \oplus \Gamma^{1,1}$$

The reason is that there's only one even unimodular Lorentzian lattice in dimensions $8n + 2$, up to isomorphism (and none in other dimensions).

So, the fact that $\mathrm{E}^8 \oplus \mathrm{E}^8$ and $\Gamma^{16}$ become isomorphic when you take their direct sum with $\Gamma^{1,1}$ is an instance of this general pattern.