**Edit:** According to essential comment of YCore I revise the question.


Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group $G$ whose differential graded algebra of all $G$-left invariant differential forms be isomorphic to $A$? This is a differential form analogy of the classical fact that " Every finite dimensional Lie algebra is the Lie algebra of a Lie group".


I browsed [this arXiv paper](https://arxiv.org/abs/1906.09630) but I did not find a result for the graded algebra case.



Is a finite dimensional graded algebra isomorphic to the equivariant de Rham complex of a Lie group?
No. A commutative differential graded algebra A is isomorphic
to the Chevalley-Eilenberg algebra of a finite-dimensional Lie algebra L
if and only if its underlying graded algebra is the exterior
algebra on A\_1, which must be finite-dimensional.
In this case we have L=(A\_1)\* and the Lie bracket on L is the dual of the differential
A\_1→A\_2.


Accordingly, any commutative dga A whose underlying commutative graded algebra
is not an exterior algebra on A\_1 is a counterexample.


