  Using generalized Tanaka-Webster connection, we considered a real
hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$
when the GTW Reeb Lie derivative of the structure Jacobi operator coincides
with the Reeb Lie derivative. Next using the method of simultaneous
diagonalization, we prove a complete classification for a real hypersurface in
$G_2({\mathbb C}^{m+2})$ satisfying such a condition. In this case, we have
proved that $M$ is an open part of a tube around a totally geodesic
$G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
