I was toying with solvable Lie algebras when I stumbled upon a fact that I can't make sense out of, and I suspect I made an error.
I will be talking about groups here, because this is a category that's more familiar to most people.
We know that abelianization has universal property: let $f: G \to A$ be a morphism from a group $G$ to an abelian group $A$. Then it factors through the abelianization $G' := G / [G, G]$ of the group $G$.: $$f: G \ \xrightarrow{\mathrm{ab}_G} \ G' \ \xrightarrow{f'} \ A.$$ Now let's consider a normal subgroup $H \lhd G$ (with the embedding $i: H \to G$) such that $G/H$ is abelian. Then the projection $p: G \to G/H$ factors through $G'$: $$p: G \ \xrightarrow{\mathrm{ab}_G} \ G' \ \xrightarrow{p'} \ G/H.$$ Then what can we say about relation between $H$ and $G'$? My intuition is that we should reverse arrows and then we get $H \leq [G, G]$, but because $p = p' \circ \mathrm{ab}_G$ we actually have $$[G, G] = \ker \mathrm{ab}_G \leq \ker p = H.$$ Did I make a mistake or is this all correct? If it's the latter, then how to make sense out of it?