Cartan-Weil model for Equivariant Cohomology Let $\mathfrak{g}$ be the Lie algebra of a Lie group $G$ which acts on a manifold $M$.
It is quite standard that the basic forms in $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ form a model for the singular equivariant cohomology of $M$. However, I have never seen a proof and it is not straightforward to me. Could someone give a sketch or a reference of the proof of this fact? It is probably in one of Cartan's papers but I haven't been able to find it.

Here goes some background:
We define its Weil algebra by $W^*(\mathfrak{g}^*)=S^*(\mathfrak{g}^*) \otimes \wedge^*(\mathfrak{g}^*)$ there is also a natural differential operator $d_W$ which makes $W*(\mathfrak{g}^*)$ into a complex. We define $d_W$ as follows:
Choose a basis $e_1,...,e_n$ for $\mathfrak{g}$ and let $e^*_1,...e^*_n$ its dual basis in $\mathfrak{g}^*$. Let $\theta_1,...,\theta_n$ be the image of $e^*_1,...e^*_n$ in $\wedge(\mathfrak{g}^*)$ and let $\Omega_1,...,\Omega_n$ be the image of $e^*_1,...e^*_n$ in $S(\mathfrak{g}^*)$. Let $c_{jk}^i$ be the structure constants of $\mathfrak{g}$, that is $[e_j,e_k]=\sum_{i=1}^nc_{jk}^ie_i$. Define $d_W$ by
\begin{eqnarray}
d_W\theta_i=\Omega_i- \frac{1}{2}\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k
\end{eqnarray}
and
\begin{eqnarray}
d_W\Omega_i=\sum_{j,k}c_{jk}^i\theta_j \Omega_k
\end{eqnarray}
and extending $d_W$ to $W(\mathfrak{g})$ as a derivation.
We can also define interior multiplication $i_X$ on $W(\mathfrak{g}^*)$ for any $X \in \mathfrak{g}$ by
\begin{eqnarray}
i_{e_r}(\theta_s)=\delta^r_s, i_{e_r}(\Omega_s)=0
\end{eqnarray}
for all $r,s=1,...,n$ and extending by linearity and as a derivation. 
Now consider $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ as a complex. Using this definition of interior multiplication, together with the usual definition of interior multiplication on forms, we define the basic complex of $\Omega^*(M) \otimes W(\mathfrak g^*)$:
We call $\alpha \in \Omega^*(M) \otimes W(\mathfrak{g}^*)$ a basic element if $i_X(\alpha)=0$ and $i_X(d \alpha)=0$. Basic elements in $\Omega^*(M) \otimes W(\mathfrak{g}^*)$ form a subcomplex which we denote by $\Omega^*_G (M)$.
The claim is that $H^*(\Omega^*_G (M))=H^*(M \times_G EG)$ where the right hand side denotes the singular equivariant cohomology of $M$.
 A: see the very nice book of Guillemin-Sternberg (Supersymmetry and ...); it also has a reprint of Cartan's paper.
A: As mentioned by the user SGP, the book Supersymmetry and Equivariant de Rham Theory by Victor W Guillemin and Shlomo Sternberg discuss about Cartan model.
 One of the intentions is to prepare the reader to understand Cartan's papers:


*

*Notions d'algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie, Colloque de Topologie, C.B.R.M., Bruxelles 15-27 (1950)

*La transgression dans un groupe de Lie et dans un espace
fibré principal, Colloque de Topologie, C.B.R.M., BruxeIles 57-71 (1950)


Preface of the book : 

This is the second volume of the Springer collection Mathematics Past
  and Present.  In the first volume, we republished Hörmander's
  fundamental papers Fourier integral operators together with a brief
  introduction written from the perspective of 1991. The composition of
  the second volume is somewhat different: the two papers of Cartan
  which are reproduced here have a total length of less than thirty
  pages, and the 220 page introduction which precedes them is intended
  not only as a commentary on these papers but as a textbook of its own,
  on a fascinating area of mathematics in which a lot of exciting
  innovation have occurred in the last few years. Thus, in this second
  volume the roles of the reprinted text and its commentary are
  reversed. The seminal ideas outlined in Cartan's two papers are taken
  as the point of departure for a full modern treatment of equivariant
  de Rham theory which does not yet exist in the literature.

Introduction : 

The  year  2000  will  be  the  fiftieth  anniversary  of  the  publication  of  Henri Cartan's  two  fundamental  papers  on  equivariant  De  Rham  theory  "Notions d'algèbre  différentielle;  applications  aux  groupes  de  Lie  et  aux  variétés  où opère  un  groupe  de  Lie"  and  "La  trangression  dans  un  groupe  de  Lie  et  dans un  espace  fibré  principal."  The  aim  of  this  monograph  is  to  give  an  updated account  of  the  material  contained  in  these  papers  and  to  describe  a  few  of the  more  exciting  developments  that  have  occUfred  in  this  area  in  the  five decades  since  their  appearance. 

