e
-\langle {\mathcal{S}}\Xi(e^{-\Phi}F\Psi e^\Phi), (e^{-\Phi}F\Psi e^\Phi)^2\rangle\,.
\label{I-1}\end{aligned}$$ Here the second term vanishes owing to (\[BPZ S\]) and (\[small to large\]): $$\begin{aligned}
- \langle {\mathcal{S}}\Xi(e^{-\Phi}F\Psi e^\Phi), (e^{-\Phi}F\Psi e^\Phi)^2\rangle\ =&\
{\langle\!\langle}(e^{-\Phi}F\Psi e^\Phi), \{(e^{-\Phi}F\Psi e^\Phi), {\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi)\}{\rangle\!\rangle}\nonumber\\
=&\
\frac{2}{3}\Big(
{\langle\!\langle}{\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi), (e^{-\Phi}F\Psi e^\Phi)^2{\rangle\!\rangle}\nonumber\\
&\hspace{10mm} 
+ {\langle\!\langle}(e^{-\Phi}F\Psi e^\Phi), \{(e^{-\Phi}F\Psi e^\Phi), {\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi)\}{\rangle\!\rangle}\Big)
\nonumber\\
=&\ 0\,.\end{aligned}$$ The first term in (\[I-1\]) can further be calculated as $$\begin{aligned}
\textrm{(I)} =&\
- \langle {\mathcal{S}}(e^{-\Phi}F\Psi e^\Phi), \widetilde{A}_Q\rangle\
=\
\langle F\Psi, e^\Phi({\mathcal{S}}\widetilde{A}_Q)e^{-\Phi}\rangle
\nonumber\\
 =&\
\langle F\Psi, QA_{\mathcal{S}}\rangle\
=\ \langle A_{\mathcal{S}}, QF\Psi\rangle\,,
\label{I}\end{aligned}$$ where we have used the relation (\[dual relation\]) with $(\mathcal{O}_1,\mathcal{O}_2)=(Q,{\mathcal{S}})$, and the identity $$\begin{aligned}
 \eta(e^{-\Phi}F\Psi e^\Phi)\ =&\ e^{-\Phi} (D_\eta F\Psi) e^\Phi\ =\ 0\,.
$$ Summing (\[II\]), (\[III\]), and (\[I\]), the variation of the action under the space-time supersymmetry transformation finally becomes $$\delta_{\mathcal{S}}S\ =\ 
\langle A_{\mathcal{S}}, \left(QF\Psi - F(Q\Psi+X\eta F\Psi)
+ F\Xi[QA_\eta+(F\Psi)^2, F\Psi]\right)\rangle\,,$$ which vanishes due to the identity (4.89) in Ref.: $\delta_{\mathcal{S}}S=0$. Hence the complete action (\[complete action\]) is invariant under the transformation (\[complete transformation\]).

Algebra of transformation {#sec algebra}
=========================

Starting from a natural linear transformation (\[linearized tf\]), we have constructed the nonlinear transformation (\[complete transformation\]) as a symmetry of the c