Let's restrict to finite dimensional case.
Functor $(-)^{\mathrm{ab}}: \mathrm{LieAlg} \to \mathrm{AbLieAlg}$ is left adjoint of the inclusion functor $i: \mathrm{AbLieAlg} \to \mathrm{LieAlg}$. Consider the same construction for inclusion of other categories of: Nilpotent Lie Algebras, Solvable Lie Algebras, Simple Lie Algebras into the category of finite dimensional Lie algebras.
The question is: What do we get as left adjoint in these cases?
I'm almost sure that in first two cases we will get factorization by limit of lower (resp. upper) central series. But for Simple Algebras maybe there is no adjoint functor.