Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism:

$$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$

Which in our case is an isomorphism since $G$ is compact. The procedure I know to prove this is pretty ad-hoc. Starting with a $G$-bundle on a manifold $M$ we pick an arbitrary connection and evaluate invariant polynomials on its curvature form to get characteristic classes.

What is the representation theoretic viepoint on the isomorphism $H^{*}(BG;\mathbb{R}) = (Sym^{\bullet} \mathfrak{g^*})^G$?

For a finite dimensional lie group (adding compact here doesn't matter) is there always a canonical way to build $BG$ as a colimit of homogeneous manifolds?

Recently i found that many computations in algebraic topology can be simplified using representation theory. Here are some more complutations I'd like to be able to understand in representation theoretic terms:

**1. Cohomology ring of a homogeneous space $H^*(G/H)$.**

**2. Cohomology ring of a parallel curvature cartan geometry $(P, \omega)$ for the pair $(\mathfrak{g},H)$ with curvature form $K \in Hom(\bigwedge^2\mathfrak{g/h}, \mathfrak{g})$.** (side question: is $K$ some kind of cocycle here?).

Is there a reference for these kind of computations?