Ellipticity of divergence and curl

Let $M$ be a 3-dimensional Riemannian manifold with Riemannian metric $g$. Let $E = S^2 T^* M$ be the vector bundle of symmetric covariant 2-tensors on $M$, and let $F = T^* M \oplus S^2 T^* M$. We consider the linear differential operator $L : \Gamma(E) \to \Gamma(F)$, where $\Gamma(B)$ denotes sections of a bundle $B$, defined by: \begin{equation} L_1(k) = ({\text{div}} k, {\text{curl}} k) \end{equation} and also the related operator \begin{equation} L_2(k) = ({\text{div}} k, \text{S}{\text{curl}} k). \end{equation} Here, the curl of a symmetric 2-covariant tensor field is defined by: $\text{curl}(k){lm} = \frac{1}{2}\epsilon{lmj}.$ We use the (nonstandard) notation Scurl to denote the symmetrized curl, which is just the symmetrization of the curl. The symmetrization of a 2-covariant tensor $T_{ab}$ is \begin{align} ST_{ab} = T_{ab} + T_{ba}. \end{align} (Note that many sources put a factor of 1/2 here.) We now investigate the principal symbols of these operators and discuss their ellipticity, i.e. when or if they have zero kernel. The principal symbol of the divergence operator in the direction $\xi \in T^*M$ is a linear map $\sigma_}(\xi) : E \to F$ defined by: \begin{equation} \sigma_}(\xi)(k)m = g^{ij}\xi_i k{jm} = k_{jm}\xi^j \end{equation}