  In this paper, first, we investigate the commuting property between the
normal Jacobi operator~${\bar R}_N$ and the structure Jacobi operator~$R_{\xi}$
for Hopf real hypersurfaces in the complex quadric~$Q^m = SO_{m+2}/SO_mSO_2$,
$m \geq 3$, which is defined by ${\bar R}_N R_{\xi} = R_{\xi}{\bar R}_N$.
Moreover, a new characterization of Hopf real hypersurfaces with $\mathfrak
A$-principal singular normal vector field in the complex quadric~$Q^{m}$ is
obtained. By virtue of this result, we can give a remarkable classification of
Hopf real hypersurfaces in the complex quadric~$Q^{m}$ with commuting Jacobi
operators.
