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This note revisits recent results regarding the geometry and moduli ofsolutions of the heterotic string on manifolds $Y$ with a $G_2$ structure. Inparticular, such heterotic $G_2$ systems can be rephrased in terms of adifferential $\check {\cal D}$ acting on a complex $\check\Omega^*(Y , {\calQ})$, where ${\cal Q}=T^*Y\oplus{\rm End}(TY)\oplus{\rm End}(V)$ and $\check{\cal D}$ is an appropriate projection of an exterior covariant derivative${\cal D}$ which satisfies an instanton condition. The infinitesimal moduli arefurther parametrised by the first cohomology $H^1_{\check {\cal D}}(Y,{\calQ})$. We proceed to restrict this system to manifolds $X$ with an $SU(3)$structure corresponding to supersymmetric compactifications to four dimensionalMinkowski space, often referred to as Strominger--Hull solutions. In doing so,we derive a new result: the Strominger-Hull system is equivalent to aparticular holomorphic Yang-Mills covariant derivative on ${\calQ}\vert_X=T^*X\oplus{\rm End}(TX)\oplus{\rm End}(V)$.

Restrictions of Heterotic $G_2$ Structures and Instanton Connections