In this paper we give a characterization of real hypersurfaces of type (A) in a complex two-plane Grassmannian $G_2(mathbb{C}^{m+2})$ which is a tube over a totally geodesic $G_2(mathbb{C}^{m+1})$ in $G_2(mathbb{C}^{m+2})$, in terms of two commuting conditions related to the normal Jacobi operator and the shape operator.