Directional elastic strain Green operator (${\tilde{\Gamma }}_C(n)=B_n(B_n^{T}C{B}_n{)}^{-1}{B}_n^T$)

Let $B_{n}a=\mathrm{sym}(a\otimes n)$; its adjoint on symmetric stress tensors is $B_n^T\tau =\tau n$. Then $B_n^{T}C{B}_n$ is the acoustic tensor. The directional elastic strain Green operator satisfies the resolvent identity ${\tilde{\Gamma }}_r(C^r-C^s){\tilde{\Gamma }}_s={\tilde{\Gamma }}_s-{\tilde{\Gamma }}_r$, by the elementary inverse identity $K_r^{-1}(K_r-K_s)K_s^{-1}=K_s^{-1}-K_r^{-1}$.