Proof of Part 3 of the Big (Subscript[𝔤, 0])-Lemma

Under [e,f]=h, h central, 𝕆((e^(βe+αf+δef))|fe)=𝕆(((νe)^(ν(-αβh+βe+αf+δef)))|ef), with ν=(((1+hδ))^(-1)).

I know about 5 proofs. Here are the most recent two; they start the same:

𝕆((e^(βe+αf+δef))|fe)=𝕆((e^(δ(∂_α)(∂_β)))