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 a commutative $R$-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).