About the Dirac operator

Question

Let $(M,g)$ be a spin manifold with Dirac operator $\cal D$, I define for a fixed vector field $X$, a proper spinor for the proper value $\lambda$ as :

$${\cal D} \psi_{\lambda}=\lambda (X.\psi_{\lambda})$$

Have we an orthonormal basis for the products $<\psi,\psi'>$ and $<X.\psi,\psi'>$ of such proper spinors?