There is a joke definition of a Lie algebra, due to my adviser John Moore, that is relevant. His definition of a Lie algebra over a commutative ring $R$ is that it is a module $L$ with a bracket operation such that there exists an associative $R$-algebra $A$ and a monomorphism $L \to A$ of $R$-modules that takes the bracket operation to the commutator in $A$. The point is to try to build in the PBW and dodge the question of which identities characterize Lie algebras. It is equivalent to the usual definition when $R$ is a field, as one sees by proving PBW using only the standard identities, but not so over a general commutative ring.
Even over a field (char $\neq 2$ for simplicity) there is an interesting contrast with the definition of a Jordan algebra. There the analogue of the commutator is $1/2 (ab + ba)$. One writes down the identities this satisfies and defines a Jordan algebra to be a vector space that satisfies the identities. But Jordan algebras do not generally embed in associative algebras (those that do are called special).