The above construction focuses on Lie algebras and on the Lie bracket, and its skewness and antisymmetry. To some degree, these properties are incidental to the construction. Consider instead some (arbitrary) algebra (not a Lie algebra) over a vector space, that is, a vector space formula_85 endowed with multiplication formula_86 that takes elements formula_87 "If" the multiplication is bilinear, then the same construction and definitions can go through. One starts by lifting formula_88 up to formula_89 so that the lifted formula_88 obeys all of the same properties that the base formula_88 does – symmetry or antisymmetry or whatever. The lifting is done "exactly" as before, starting with