Let $(\Sigma,g)$ be a Riemannian manifold. For a differential form $\alpha$, given $d^{*}\alpha=0$, where $d^*$ is the codifferential with respect to $g$, can we rewrite the equation $d^*d\alpha=0$ as a divergence-form strongly elliptic system of equations using local coordinates (just like the case when $\alpha$ is a function) ?

My purpose is to improve the regularity of $\alpha$ given that $\alpha\in H^{-1}$ in case $g$ is not smooth (in some Sobolev space). We have $\Delta \alpha=0$, but in local coordinates this is an elliptic system of non-divergence form, which is not convenient to apply regularity results as in the function case. That's why I want to see if it's possible to rewrite the condition $d^*\alpha=0$ and rewrite it as a divergence-form system.