  In this paper, we introduce a new commuting condition between the structure
Jacobi operator and symmetric (1,1)-type tensor field $T$, that is,
$R_{\xi}\phi T=TR_{\xi}\phi$, where $T=A$ or $T=S$ for Hopf hypersurfaces in
complex hyperbolic two-plane Grassmannians. By using simultaneous
diagonalzation for commuting symmetric operators, we give a complete
classification of real hypersurfaces in complex hyperbolic two-plane
Grassmannians with commuting condition respectively.
