Let $(\Sigma,g)$ be an $n$-dimensional 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 as much as possible the regularity of $\alpha$ given that $\alpha\in L^2_{loc}$ in case $g\in W_{loc}^{1,n}$ is not continuous. If $\alpha$ is a function, then we can prove $\alpha\in W^{2,p}_{loc}$ for any $p<n$. If $\alpha$ is a $k$-form for $k\ge 1$ 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 insert the condition $d^*\alpha=0$ to $d^*d\alpha=0$ to get a divergence-form system.