the dual of the Weil algebra .
In the first formulation this may be identified with the dg-Lie algebra whose
elements in degree -1 are the contractions with ;
elements in degree 0 are the inner derivations ;
the differential is given by the commutator ;
the bracket is the graded commutator bracket of derivations:
So this is the full subalgebra of the automorphism ∞-Lie algebra of on the inner derivations.
See Weil algebra as CE-algebra of inner derivations for more details.
The first formulation makes manifest that is the structure that has historically been called Cartan calculus.
The structure of is of course in itself very simple and goes as such back at least to Cartan.