  In this paper, we have considered a new commuting condition, that is,
$(R_\xi\phi) S = S (R_\xi\phi)$ \big(resp. $(\Bar{R}_N\phi) S = S
(\Bar{R}_N\phi$)\big) between the restricted Jacobi operator~$R_\xi\phi$ (resp.
$\Bar{R}_N\phi$), and the Ricci tensor $S$ for real hypersurfaces $M$ in
$G_2({\mathbb C}^{m+2})$. In terms of this condition we give a complete
classification for Hopf hypersurfaces $M$ in $G_2({\mathbb C}^{m+2})$.
