*I'll be using homological grading throughout this question.*

Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods:

$$H^{\bullet}(BG;\mathbb{R}) \cong Sym^{\bullet}(\mathfrak{g}^*[2])^G$$

$$H^{\bullet}(G; \mathbb{R}) \cong Sym^{\bullet}(\mathfrak{g}^*[1] )^G$$

Where on the RHS we take invariants w.r.t. the adjoint action of $G$ on the lie algebra. Replacing $G$ with its complexification (which is the complex points $\mathbb{G} (\mathbb{C})$ of a connected semi-simple complex algebraic group) doesn't change the cohomology so let us do this. Next let's interpret the graded rings above as (derived if you like) prestacks (functors $CommAlg_{\mathbb{C}} \to \infty$-groupoids). Translating to this language we get the following (which is just a different way to notate the above):

$$SpecH^{\bullet}(B\mathbb{G}(\mathbb{C});\mathbb{C}) \cong \mathfrak{g}_{\mathbb{C}}[-2] \text{ //}G \cong \mathbb{T}_{B\mathbb{G}}[-1] \text{ //} \mathbb{G}$$

$$Spec H^{\bullet}(\mathbb{G}(\mathbb{C}); \mathbb{C}) \cong \mathfrak{g}_{\mathbb{C}}[-1] \text{ //}G \cong \mathbb{T}_{B\mathbb{G}} \text{ //} \mathbb{G}$$

I find it weird that both sides of the equation naturally take as input a connected semisimple complex algebraic group $\mathbb{G}$ but the isomorphism itself is of a transcendental nature (by passing through the real points of the compact real form as a topological group). This mysterious relationship does not manifest itself at the level of the proofs of these statement (at least in the proofs I know). It would be nice to know whether these interpretations can be turned to proofs (at least giving one of the results given the other one):

Question 1:Is their ahigher mechanismat work here? Phrased differently: Are these isomorphisms a formal consequences of the formalism of DAG together with some basic facts about groups?