Pre-supersoluble Lie algebra (PSS)

Call a Lie algebra pre-supersoluble if it is abelian or its quotient by its centre has a one-dimensional ideal. Every nonzero such algebra has nonzero abelianization over every field. To prove this, choose $x$ outside the centre with $Z(L)+Fx$ an ideal and write $[a,x]=\lambda (a)x+z(a)$ with $z(a)$ central. The Jacobi identity gives $\lambda ([a,b])=0$. If $\lambda \ne 0$, the derived algebra lies in its proper kernel. If $\lambda =0$, Jacobi gives $[[a,b],x]=0$, so the derived algebra lies in the proper centralizer of an element of a Lie algebra of $x$. The abelian case is immediate. This condition does not imply solvability: ${\mathfrak{sl}}_2(\mathbb{C})$ direct sum the affine Lie algebra of the line has zero centre and a one-dimensional ideal in the affine factor, but is not solvable.