# Show that integral lattice is even ($E_8$)

Consider the lattice $$\Gamma_8$$ consisting of the points $$\vec{a} = (a_1,\ldots,a_8) \in \mathbb{R}^8$$ satisfying $$a_i-a_j \in \mathbb{Z}$$, $$2 a_i \in \mathbb{Z}$$ and $$\sum_{i=1}^8 a_i \in 2\mathbb{Z}$$, with the inner product $$\vec{a} \circ \vec{a}' = \vec{a} \cdot \vec{a}' = \sum_i a_i^2$$. Show that $$\Gamma_8$$ is even.

I have boiled this down to the following proof: Suppose that I have a list of eight half integers $$(\frac{a_1}{2},\ldots,\frac{a_8}{2})$$ that sum to an even number, show that the sum of their squares $$(\frac{a_1^2}{4},\ldots,\frac{a_8^2}{4})$$ is an even number. I am convinced that this is true but the proof eludes me.

• Odd squares are congruent to $1$ modulo $8$ – Lord Shark the Unknown Apr 29 at 18:32