= Solution
The equilibrium quantities $\rho,p,\mathbf B,\Phi$ are real, and the gravitational <Hessian> $\Phi_{,ij}=\partial_i\partial_j\Phi$ is symmetric. The crucial <tensor> property is
$$
\boxed{V_{ijkl}=V_{klij}.}
$$
This follows directly from the displayed expression for $V$: the products $\delta_{ij}\delta_{kl}$ and $\delta_{il}\delta_{jk}$ are unchanged by interchanging the pairs $(i,j)$ and $(k,l)$; $B_jB_l\delta_{ik}$ is also unchanged; and the two terms involving $B_iB_j\delta_{kl}$ and $B_kB_l\delta_{ij}$ exchange with each other. All coefficients are real.
Multiply $F\boldsymbol\xi$ by $\rho\boldsymbol\eta^*$ and integrate by parts over all space. For admissible <displacement> fields, the boundary contribution at infinity vanishes: the fluid <pressure> and density are confined to the body, and the magnetic stress decays in the surrounding conducting medium. The conducting exterior is included in the integration. For a piecewise description, the usual <displacement> and traction matching conditions cancel the two opposite interface contributions; alternatively, take a smooth transition and then its limit. Thus
$$
\langle\boldsymbol\eta,F\boldsymbol\xi\rangle=-\int\rho\eta_i^*\Phi_{,ij}\xi_j\,dV-\int(\partial_j\eta_i^*)V_{ijkl}(\partial_l\xi_k)\,dV.
$$
The first term is a Hermitian pairing because $\Phi_{,ij}=\Phi_{,ji}$, and the second is Hermitian because $V_{ijkl}=V_{klij}$. Taking the complex conjugate of the same expression with $\boldsymbol\eta$ and $\boldsymbol\xi$ interchanged and relabelling $(i,j)\leftrightarrow(k,l)$ reproduces it. Hence
$$
\boxed{\langle\boldsymbol\eta,F\boldsymbol\xi\rangle=\langle F\boldsymbol\eta,\boldsymbol\xi\rangle.}
$$
This establishes the required <self-adjoint operator> identity in the mass-weighted <inner product>, on the physical <displacement> domain with the stated decay and interface conditions. The negligible-density exterior can be treated as a small-positive-density conducting limit when defining this weighted pairing; its magnetic contribution is retained even as its inertia becomes negligible.
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