Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it.
$G$ is a group, $\mathbb{Z}[G]$ its integral group ring, $I_G$ the augmentation ideal (i.e. the kernel of the map $\mathbb{Z}[G]\rightarrow\mathbb{Z}$ sending all group elements to 1), and $G/G'$ is the abelianization of $G$. I'm attempting to show that $I_G/I_G^2$ is isomorphic to $G/G'$.
It's straightforward to show that $$(g-1)+(h-1)\equiv(gh-1)\bmod I_G^2\hskip0.5in(*)$$ and thus every equivalence class mod $I_G^2$ is equal to some $(g-1)+I_G^2$.
Now all we have to do is define a map $\phi:G/G'\rightarrow I_G/I_G^2$ and show that it has an inverse $\psi:I_G/I_G^2\rightarrow G/G'$. The definitions are obvious enough: $$\phi(gG')=(g-1)+I_G^2$$ $$\psi((g-1)+I_G^2)=gG'$$ and starred equation shows that these are homomorphisms. But! My problem is showing that these maps are well-defined. For example, if $(g-1)+I_G^2=(h-1)+I_G^2$, i.e. $$(g-1)-(h-1)\equiv (gh^{-1}-1)\equiv0\bmod I_G^2,$$ we need to show that $$\psi((g-1)+I_G^2)=gG'=hG'=\psi((h-1)+I_G^2),$$ i.e. $gh^{-1}\in G'$.
If we have $gh^{-1}\in G'$, i.e. $gh^{-1}$ equals some $\prod_{a,b\in G} (aba^{-1}b^{-1})^{n_{a,b}}$, then $$gh^{-1}-1=\left(\prod_{a,b\in G} (aba^{-1}b^{-1})^{n_{a,b}}\right)-1,$$ and, unwinding using the starred equation, $$\left(\prod_{a,b\in G} (aba^{-1}b^{-1})^{n_{a,b}}\right)-1\equiv \sum n_{a,b}(aba^{-1}b^{-1}-1)\equiv$$ $$\sum n_{a,b}\left[(ab-1)-(a-1)-(b-1)\right]=\sum n_{a,b}(a-1)(b-1)\equiv0\bmod I_G^2$$ but for some reason I can't make this work the other way. I'm sure I'm being silly; someone please point out where.