= Pre-supersoluble Lie algebra
{title2=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 $\lambda\ne0$, 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.
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