This question is a duplicate of that 2010 MO question.

I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$.
Clearly, any integral representation of $C_2$ is a direct sum of *indecomposable* integral representations.

The following result is well-known:

Theorem.The group $C_2$ has exactly 3 isomorphism classes of indecomposable integral representations:(1) trivial;

(2) the sign representation;

(3) the 2-dimensional representation with matrix $\left(\begin{smallmatrix}0 & 1\\ 1 & 0\end{smallmatrix}\right).$

This result was stated in the answer of Victor Protsak. See also the answer of Todd Leason.

In his comment Victor Protsak gives a reference. He writes: "Curtis and Reiner, Chapter 11. It's a special case of a theorem in Section 74 which classifies integral representations of cyclic groups of prime order. Naturally, this case is much easier and can be done by hand."

Question.How to prove the above theorem "by hand", without reference to the book by Curtis and Reiner?

**Motivation:** I am working now with algebraic $\mathbb R$-tori.
They are classified by integral representations of the Galois group
${\rm Gal}({\mathbb C}/{\mathbb R})$, which is a group of order $2$.
In order to understand the well-known classification of indecomposable $\mathbb R$-tori,
I need to understand the well-known classification
of indecomposable integral representations of ${\rm Gal}({\mathbb C}/{\mathbb R})$.

I asked this seemingly elementary question on Mathematics StackExchange, but got no answers or comments, so I ask it here.