of the implementation of the BRST algorithm can be found in [@BurCapDre:]. All dummy variables are noted as $d_i$ where $i$ is an integer. Unless stated otherwise, there is an implicit summation on all dummy variables. For the examples in which the Jacobi identity is not trivially satisfied, the expressions have been normalized so that no combinations of terms in a polynomial belongs to the polynomial ideal generated by the left hand side of the Jacobi identity. In particular, polynomials in this ideal are represented by identically null expressions.

Self-Reproducing Algebras
-------------------------

As we have just defined, the basic Poisson brackets for the generators $G_d$ of the [*self-reproducing algebra*]{} are given by $$[G_{d_1}, G_{d_2}] = M_{d_1 d_2} G_{d_1} G_{d_2}$$ without summation over the dummy variables $d_1$ and $d_2$. The matrix $M$ is antisymmetric, but otherwise arbitrary.

The seven first orders of the covariant BRST charge are given by $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}=
\[G_{d_{1}}\,\eta^{d_{1}}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}=
\[\frac{G_{d_{1}}\,M_{d_{1}d_{2}}\,\eta^{d_{1}}\,
        \eta^{d_{2}}\,{{\cal P}}_{d_{2}}}{
        2}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (2)} \\ \Omega \end{array}}}=
\[\frac{-
        \(G_{d_{1}}\,M_{d_{1}d_{2}}\,\eta^{d_{1}}\,
          \eta^{d_{2}}\,\eta^{d_{3}}\,{{\cal P}}_{d_{2}}\,
          {{\cal P}}_{d_{3}}\,
          \(M_{d_{1}d_{3}}
            +M_{d_{2}d_{3}}
          \)
        \)
        }{
        12}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7}     \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (3)} 