  We establish a decoupling result for the $P$ and $S$ waves of linear,
isotropic elasticity, in the setting of twice-differentiable Lam\'e parameters.
Precisely, we show that the $P\leftrightarrow S$ components of the wave
propagation operator are regularizing of order one on $L^2$ data, by
establishing the diagonalization of the elastic system modulo a $L^2$-bounded
operator. Effecting the diagonalization in the setting of twice-differentiable
coefficients depends upon the symbol of the conjugation operator having a
particular structure.
