Let $ (S, \{. \}, [.]) $ be an algebra over a vector space endowed with two bilinear maps $ \{. \}, [.] : S \times S \rightarrow S$ and satisfying some compatibility conditions.

In example, if S is a Poisson algebra; the structure of Poisson module over S is well known

im asking if there is ( in a abstract way) a general definition of module over $S$ compatible with the two structures $\{. \}, [.] $

Edited; the example i am interested in, is when $S$ is a CPA-algebra (link); that is $ (S, [.]) $ is a Lie algebra and $ (S,.) $ a commutative algebra such that $$[x,y].z = x.(y.z) - y.(x.z)$$ $$x.[y,z] = [x.y,z] + [y,x.z]$$

bimodule because of the antisymmetry of the bracket and symmetry of multiplication. If you accept that, then there is a relatively natural definition: a module $E$ is a vector space (over the same underlying field) such that $S \oplus E$ is the "same kind" of algebra (i.e. it satisfies the same compatibility conditions), where you set $\{e_1, e_2\} = [e_1, e_2] = 0$ for $e_1, e_2 \in E$. $\endgroup$