We establish a decoupling result for the P and S waves of linear, isotropic elasticity, in the setting of twicedifferentiable Lamé parameters. Precisely, we show that the P ↔ 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.