\begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}} + (-)^{k+1} {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_2 \end{array}})\end{aligned}$$ where $$\begin{aligned}
{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}} &=& G_1 \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^1
                 {{\cal P}}_{\alpha_1} \cdots {{\cal P}}_{\alpha_k} \\
{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_2 \end{array}} &=& G_{\alpha_k} \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^1
                 {{\cal P}}_1 {{\cal P}}_{\alpha_1} \cdots {{\cal P}}_{\alpha_{k-1}}\end{aligned}$$ and $$\begin{aligned}
\alpha_1 &=& -1/2, \; \alpha_2 = -1/12, \; \alpha_3 = 0 \\
\alpha_k &=& -\frac{1}{k+1} \sum_{l=1}^{k-3} \alpha_{l+1} \alpha_{k-l-1}
            \quad \mbox{for $k > 3$}\label{eq:AlpRec}.\end{aligned}$$

This can be seen as the only non zero brackets involved in the construction of the BRST charge are $$\begin{aligned}
[{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}}, {{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]_{\mbox{orig}} &=& (-)^{k+1}{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} \\{}
[{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}}, {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (l)} \\ T_2 \end{array}}]_{\eta{{\cal P}}} &=&
  (-)^{l(k+1)}{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+l-1)} \\ S \end{array}},\end{aligned}$$ where $${\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} = G_1 G_\alpha \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^\alpha
\eta^1
                 {{\cal P}}_{\alpha_1} \cdots {{\cal P}}{\alpha_k}.$$ We further have $$\label{eq:sigmaS}
\sigma {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} = (-)^{k+1} {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ T_2 \end{array}} - {\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ T_1 \end{array}}$$ Given these relations, 