= Solution
Use the same real <elastic stiffness tensor> for both fields, including its minor and major symmetries, and let $t_i^a=\widehat\sigma_{ij}^a n_j$, with outward unit boundary normal $n_j$. No <complex conjugation> is needed for the initial bilinear identity. Define
$$
F_j=\widehat\sigma_{ij}^a\widehat u_i^b-\widehat u_i^a\widehat\sigma_{ij}^b.
$$
The <divergence> has two force-balance terms and two cross-gradient terms. The latter cancel because
$$
\widehat\sigma_{ij}^a\partial_j\widehat u_i^b=\widehat\sigma_{ij}^b\partial_j\widehat u_i^a;
$$
this follows by exchanging the two strain-gradient factors using major symmetry and the minor pair symmetries. The frequency-domain equations then give
$$
\partial_jF_j=-(\rho\omega_a^2\widehat u_i^a+\widehat f_i^a)\widehat u_i^b+\widehat u_i^a(\rho\omega_b^2\widehat u_i^b+\widehat f_i^b).
$$
Integrating and applying the <divergence theorem> proves
$$
\boxed{\int_{\mathcal D}\left[-(\rho\omega_a^2\widehat u_i^a+\widehat f_i^a)\widehat u_i^b+\widehat u_i^a(\rho\omega_b^2\widehat u_i^b+\widehat f_i^b)\right]dV=\int_{\partial\mathcal D}(t_i^a\widehat u_i^b-\widehat u_i^at_i^b)dS.}
$$
This is the <Betti identity for elastodynamic fields>. Spatially varying stiffness is allowed because its <derivatives> remain inside the <divergence>. The original PDF's constitutive contraction is $c_{ijpq}\partial_p\widehat u_q$; the TeX's repeated <displacement field> index is erroneous.
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