In his seminal 2004 paper "Higher Composition Laws I" in the *Annals of Mathematics*, (doi:10.4007/annals.2004.159.217), Bhargava proves that for fixed $D \neq 0$, there is a bijective correspondence between the following two collections:

- Primitive Bhargava cubes of discriminant $D$ modulo the action of $(\mathrm{SL}_2(\mathbb{Z}))^{\times 3}$, and
- Triplets of oriented ideal classes $I_1, I_2, I_3$ in the narrow class group of the unique oriented quadratic ring of discriminant $D$ with the property that $I_1 \cdot I_2 \cdot I_3$ is the class of the unit ideal.

He then claims that, as a corollary of the above correspondence, one recovers the standard bijective correspondence due to Gauss between primitive integral binary quadratic forms and ideal classes in the narrow class group. Why is this true, that Gauss's correspondence is a corollary? I don't immediately see how this follows.

What I know: It's not hard to see explicitly that there is a Bhargava cube such that one of its three associated quadratic forms equals any given integral binary quadratic form. Thus, the map sending an ideal class to its norm form is surjective. I'm having trouble showing that this map is injective.