Given a divided power ring $(A,I,\gamma)$ and an $(A,I)$ algebra $(B,J)$ we can consider the divided power envelope. I want to show : $$D_{B[x_i], \gamma}(JB[x_i] + (x_i)) = D_{B, \gamma}(J) \langle x_i \rangle \ (*)$$ and for this I want to use Lemma 55.2.4, as indicated in Lemma 55.2.5. Unfortunately I can't figure out how to use the lemma exactly.
My thoughts: With the notation of Lemma 55.2.4. my divided power algebra will be $(D_{B[x_i],\gamma}(JB[x_i]),\bar{J},\bar{\gamma})$ and I want to consider the envelope relative to $\bar{J}+(x_i)$ (the image of $(x_i)$ in $D_{B[x_i],\gamma}(JB[x_i])$ to be precise). Then I get the following map by Lemma 55.2.4. $$ D_{B[x_i],\gamma}(JB[x_i])\langle y_i \rangle/(x_i-y_i)\to D(D_{B[x_i],\gamma}(JB[x_i]),\bar{J}+(x_i),\tilde{\gamma})$$ Then the LHS might be isomorphic to the RHS of $(*)$ and the RHS will be isomorphic to the LHS of $(*)$.
But it is not obvious to me, if this is really true and if my attempt is correct at all.