Maximal even sub-lattice in $\mathbb{Z}^n$ (reference request)

Question

Lemma: The maximal even sub-lattice in $\mathbb{Z^n}$ is $$ \Big\{ (x_1,\cdots,x_n)\in \mathbb{Z}^n ~~\big|~~ \sum_{1 \le i \le n} x_i \in \mathbb{2Z} \Big\} $$

I found the above lemma in an article, but without a proof. I would like to ask for a proof or a reference which hopefully includes a proof. Thanks.

Answer 1

It's not hard to prove it yourself. First show that it's even (easy) and then that it's maximal. For the latter, if it's not maximal then there exists a new even lattice ($L_2$) that includes your lattice $L$. Since it is genuinely bigger than $L$ then it must include a vector $a_n\in\mathbb Z^n$ with $\sum a_n=$odd. You can then consider the inner product of $a$ with a few vectors in $L\subset L_2$ to see that $L_2$ is not even, reaching a contradiction.