In this paper, from the property of Killing for structure Jacobi tensor
$\mathbb {R}_{\xi }$
, we introduce a new notion of cyclic parallelism of structure Jacobi operator
$R_{\xi }$
on real hypersurfaces in the complex two-plane Grassmannians. By virtue of geodesic curves, we can give the equivalent relation between cyclic parallelism of
$R_{\xi }$
and Killing property of
$\mathbb {R}_{\xi }$
. Then, we classify all Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in complex two-plane Grassmannians.