Suppose we define the total angular momentum as F = I+J where I is the nuclear spin angular momentum and J is the total electronic angular momentum. mF and mJ are the respective magnetic quantum numbers. The relevant F selection rules are delta_mF = 0, 1 and -1, delta_F = 0, 1 and -1. And similarly, the J selection rules are delta_mJ = 0, 1 and -1, delta_J = 0, 1 and -1. My question is, when considering a transition involving nuclear spin, in addition to satisfying the delta_F and delta_mF selection rules, do we have to also satisfy the delta_J and delta_mJ selection rules at the same time? A specific example involving two successive transitions is attached. The transitions satisfy the delta_F and delta_mF selection rules but violate the delta_mJ selection rules. Would such a two step process be allowed or forbidden? I'd appreciate any insight. Thank you!