This question is a duplicate of
[that 2010 MO question](https://mathoverflow.net/questions/27129/involutions-in-gl-nz/27145#27145).


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](https://mathoverflow.net/a/27145/4149).
See also [the answer of Todd Leason](https://mathoverflow.net/a/204589/4149).


In [his comment](https://mathoverflow.net/questions/27129/involutions-in-gl-nz/27145#comment57961_27145)
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](https://math.stackexchange.com/q/3789999/37763),
but got no answers or comments, so I ask it here.



Indecomposable integral representations of a group of order 2 "by hand"
In [Computing with real tori](https://www.math.ubc.ca/%7Ecass/research/pdf/realtori.pdf), Casselman has a nice write-up of this theorem from the point of view of not just proving that these are the only indecomposable tori, but, supposing you are given an explicit integral representation of $\operatorname C\_2$, explicitly finding/computing its decomposition into these three representations.


In fact, if you (you the general reader, not necessarily @MikhailBorovoi) aren't familiar with Bill Casselman's recent work, it's well worth checking out his page [http://www.math.ubc.ca/~cass](http://www.math.ubc.ca/%7Ecass); he's been very interested for a while in doing actual computations, in the sense of things that can be fed into a computer, relating to algebraic groups. The above is one example; others can be found at [http://www.math.ubc.ca/~cass/research/publications.html](http://www.math.ubc.ca/%7Ecass/research/publications.html), including, for example, [The computation of structure constants according to Jacques Tits](https://www.math.ubc.ca/%7Ecass/research/pdf/Tits.pdf)—things that we all know *can* be done but that most of us (at least I!) would shrink from actually *doing*, here laid out in a way that demonstrates how to carry it out practically.


(There's also some nice stuff on [mathematical graphics](http://www.math.ubc.ca/%7Ecass/graphics/)!)


