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.

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

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