Q: Solvable Lie algebras. derived algebras and ideals Let L be a vector subspace codimension $1$ in $g/ g'$, where g is a Lie algebra solvable and $g'=[g,g]$ derived algebra.
Have:

*

*$g/g'$ is abelian

In fact, if $x, y \in g$
$$[x+g',y+g']=[x,y]+g'= g'=0+g'$$
because $[x,y]\in [g,g]=g'$.


*Then L is a ideal in $g/g'$
I need help the second case.

A: Since $g/g'$ is abelian every vector subspace $I$ is an ideal. Since for every $x\in g/g', i\in I$ we have $[x,i]=0\in I$.
