Solvable Lie algebras. derived algebras and ideals

Question

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:

1. $g/g'$ is abelian

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

2. Then L is a ideal in $g/g'$

I need help the second case.

Answer 1

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