\,D_{d_{17}d_{4}}^{
           d_{18}d_{11}}\,D_{d_{18}d_{3}}^{
                             d_{19}d_{10}}\,
      D_{d_{19}d_{2}}^{
         d_{15}d_{9}}\,G_{d_{16}}
      +16\,D_{d_{6}d_{7}}^{
              d_{16}d_{13}}\,D_{d_{14}d_{5}}^{
                                d_{15}d_{12}}\nl
      \off{3827636}
      \,D_{d_{16}d_{4}}^{
           d_{17}d_{11}}\,D_{d_{17}d_{3}}^{
                             d_{19}d_{10}}\,
      D_{d_{18}d_{1}}^{
         d_{14}d_{8}}\,D_{d_{19}d_{2}}^{
                          d_{18}d_{9}}\,G_{d_{15}}
    \)
  \)
  /15120
\]
\Nl}$$

Again, these expressions are not particularly illuminating. The point emphasized here is that the calculation of the BRST charge is purely algorithmic and follows a general, well-established pattern.

Since homogeneous quadratic algebras contain the self-reproducing algebras as special case, they are generically of maximal covariant rank. More on this in [@ADresse3].

L-T algebras
------------

We now consider adding a linear term to the quadratic algebra above. The basic Poisson brackets for the generators $G_d$ are given by $$[G_{d_1}, G_{d_2}] = C_{d_1 d_2}^{d_3}  \, G_{d_3} +
   D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4}$$ where $ C_{d_1 d_2}^{d_3} $ and $D_{d_1 d_2}^{d_3 d_4}$ are antisymmetric in $d_1, d_2$, and $D_{d_1 d_2}^{d_3 d_4}$ is symmetric in $d_3, d_4$. A particular instance of such an algebra is given by Zamolodchikov algebras [@Zam:]. We will start with a specific example, and consider general quadratically nonlinear Poisson algebras next.

The generators in the example are assumed to split into $L_{a}$ and $T_b$, $a = 1, \ldots n_1$, $b = n_1 + 1, \ldots, n$, with the brackets $$\begin{aligned}
[L_{a_1}, L_{a_2}] &=& \tilde{C}_{a_1 a_2}^{a_3} L_{a_3} \nonumber \\{}
[L_{a_1}, T_{b_1}] 