Let y be any element of H and let x be any element of R. Since xy lies in H, it is nilpotent. Some power of xy = 0. For notational convenience, let u = xy, and note that 1-u is invertible, with inverse 1+u+u2+u3…+un, where un drops to 0. This means 1-xy is left invertible, and by condition (2), y is in jac(R), and H is in jac(R).
If H is a right ideal, run the same proof for 1-yx being right invertible.
All the nilpotent ideals lie in jac(R).
The same proof works if H is a nil (left) ideal. Remember that a nil ideal consists of nilpotent elements, even though the entire ideal may not be nilpotent.