The formula G(PWX((PWX a) → ¬b)) is translated first to a safety automaton:

G(PWX((PWX a) → ¬b))
=
AUTO(exists,{q1,q2},q1∧q2,(Xq1 ↔ a)∧(Xq2 ↔ (q1 → ¬b)), Gq2)

That far that good, I guess. Next, we have to convert this safety automaton to a FG-automaton which is done as explained on slide 60 of the omega-automata chapter. That will introduce a new state variable q3. There is no PG used in that formula.

L1 = G(a → X¬b) and L2 = F(a → [a WU ¬b]). Note first that we can rewrite L2 as (F ¬a) | F[a WU ¬b].

Assume L1 holds, i.e., whenever a holds, then b is false next time. In a first case, we never have a, but then we certainly have F ¬a, and thus also L2. Second, if at some point of time, we have a, then by L1, we must also have X¬b there, thus, ¬b at the next point of time. This also implies L2 since at the next point of time where ¬b holds, we also have [a WU ¬b], and thus also F[a WU ¬b].