Characterization of the exterior derivative This is a cross-post of someone else's question.

I am cross-posting this question from MSE since it hasn't received any answers.

In the paper Natural Operations on Differential Forms, the author R. Palais shows that the exterior derivative $d$ is characterized as the unique "natural" linear map from $\Phi^p$ to $\Phi^{p+1}$ (Palais' $\Phi^p$ is what is perhaps more commonly written as $\Omega^p$, and "commutes with all diffeomorphisms", I believe, means $f^*(d\omega) = d(f^*\omega)$):

the exterior derivative on $p$-forms is determined to within a scalar factor by the condition that it be a linear mapping into $p+1$ forms which commutes with all diffeomorphisms. 

I've tried to read the proof in the paper, but I'm struggling to follow the details and missing a sense of the big picture of the proof.
I'm interested in Palais' claim because this characterization is the most compelling one I have seen $-$ it seems far more appropriate as an axiomatic definition of $d$ than the definitions found in many textbooks, which often define $d$ based on properties such as
$d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$ (where $\alpha$ is a $k$-form),
$d^2=0$,
or $d(\sum \omega_{i_1...i_k}dx_{i_1} \wedge \cdots \wedge dx_{i_k}) = \sum d\omega_{i_1...i_k} \wedge dx_{i_1} \wedge \cdots \wedge dx_{i_k}$.
While these are indeed quite basic properties of $d$, they are more appropriate as theorems than as a priori assumptions. (Of course, what is more "natural" is a matter of opinion, so please don't belabor the issue.)
Since it's such an innocent-looking and natural characterization, I would like to see a clear, motivated, reasonably elementary proof of it. Why ought it to be true that there is only one natural linear map from $\Omega^p$ to $\Omega^{p+1}$, up to a constant multiple? What are the key steps of a proof?
 A: I will try to answer this question in more detail after looking back at my paper of long ago.  If anyone would like a copy of the paper they can download the pdf version at the following link: 
http://vmm.math.uci.edu/PalaisPapers/NaturalOperationsOnDifferentialForms.pdf
It would help me if I had a better idea of what steps in my argument are not clear.
Dick Palais

A: Here is a sketch of one possibility how to prove this:
The first key step is to see that the operator you are looking at has to be a differential operator. One usual way to ensure this is to require that it commutes with local diffeomorphisms (which include open embeddings). This implies that the operator is local and hence a differential operator by the Peetre theorem. 
Having this at hand, the operator has to be induced by a vector bundle map $J^k\Lambda^pT^*M\to\Lambda^{p+1}T^*M$, where $J^k$ is the $k$th jet prolongation, and this bundle map has to be compatible with the natural action of all local diffeomorphisms. Such bundle maps are then determined by linear maps between the standard fibers of these bundles, which are equviariant for the actions of the so-called jet groups. (Formally, this is proved via higher order frame bundles.) In this case, the relevant group will be $G^k_n$, where $n$ is the dimension of $M$. This is an extension of $GL(n,\mathbb R)$, formally defined as the group of $k$-jets at $0\in\mathbb R^n$ of local diffeomorphisms of $\mathbb R^n$ fixing $0$. 
Now on the target space $\Lambda^{p+1}\mathbb R^{n*}$, the jet group acts just via the usual action of $GL(n,\mathbb R)$ and this is an irreducible representation. On the other hand, restricted to the subgroup $GL(n,\mathbb R)$ the representation inducing $J^k\Lambda^pT^*M$ is the direct sum of the representations $S^{\ell}R^{n*}\otimes\Lambda^p\mathbb R^{n*}$ for $\ell=0,\dots,k$. Now there is only one summand which contains a copy of $\Lambda^{p+1}\mathbb R^{n*}$, namely the one for $\ell=1$. Hence your operator must be of first order, and its principal symbol must be the one of the exterior derivative (up to a constant multiple). 
Hence you have a multiple of the exteiror derivative up to adding an operator of order zero, i.e. a bundle map $\Lambda^pT^*M\to\Lambda^{p+1}T^*M$ which commutes with the action of all diffeomorphisms. Representation theory of $GL(n,\mathbb R)$ immediately implies that such a bundle map does not exist. 
There are lots of examples like this one discussed in the book Natural operations in differential geometry by Kolar, Michor and Slovak (MR 94a:58004) 
