Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra satisfying the additional polynomial identity $$ [[X,Y],[U,V]] = 0,\, \forall X,Y,U,V \in \mathfrak{g}. $$ I would like to learn and to know about all the polynomial identities that are satisfied by the Lie algebra $\mathfrak{g}$. Is there a finite basis for the polynomial identities of $\mathfrak{g}$ ? (Specht’s problem)
Here, finite basis is in the PI-algebra theory sense.
References
Claudio Procesi: What happened to PI theory? (arXiv:1403.5673 [math.RA])
It's a classical result of Kovacs and Newman that a metabelian nilpotent variety of Lie algebras (over a field $K$ of characteristic zero) always has the form $\mathfrak{A}^2\cap\mathfrak{N}_c$.
In other words: let $\mathcal{V}$ be a non-empty collection of Lie $K$-algebras, stable under taking arbitrary (possibly infinite) direct products, subalgebras, quotients, and consisting of metabelian nilpotent Lie algebras. Then there exists $c$ such that it exactly consists of the $c$-nilpotent metabelian Lie algebras.
So the answer is yes, and you even have a basis with two elements (the double commutator and the $(c+1)$-iterated commutator.
The basic lemma underlying this result is that if $\mathfrak{f}=\bigoplus_{k\ge 1}\mathfrak{f}_k$ is the free metabelian Lie algebra on $n$ generators, with its canonical grading, then the representation of $\mathrm{GL}_n$ on $\mathfrak{f}_k$ is irreducible for all $k$.