{x_1}\, {x_3}\,
{x_6}\,{x_7}
    \nonumber\\
  &&- 6 \,{x_2}\,{x_3}\,{x_6}\,{x_7}+ 6 \,{x_4}\,{x_5}\,{x_6}\,{x_7} +
   {{x_1}}^2\, {{x_7}}^2 -
  2\, {\sqrt{3}}\, {x_1}\, {x_2}\, {{x_7}}^2 \nonumber\\
  &&+ 3\, {{x_2}}^2\, {{x_7}}^2 +
  3\, {{x_3}}^2\, {{x_7}}^2 + 3\, {{x_5}}^2\, {{x_7}}^2 -
  2\, {\sqrt{3}}\, {x_1}\, {x_4}\, {x_6}\, {x_8}   \nonumber\\
  && +  6\, {x_2}\, {x_4}\, {x_6}\, {x_8}+6\, {x_3}\, {x_5}\, {x_6}\,
  {x_8} +     6\, {x_3}\, {x_4}\, {x_7}\, {x_8} +
  4\, {\sqrt{3}}\, {x_1}\, {x_5}\, {x_7}\, {x_8}   \nonumber\\
  &&+   {{x_1}}^2\, {{x_8}}^2
  +2\,{\sqrt{3}}\, {x_1}\, {x_2}\, {{x_8}^2}+3\, {{x_2}}^2\, {{x_8}}^2 +
  3\, {{x_4}}^2\, {{x_8}}^2   +3\, {{x_5}}^2\, {{x_8}}^2\;.\nonumber
  \end{aligned}$$

The corresponding $\widehat P $-matrix elements can be easily calculated from their definition [(\[matP\])]{}: $$\begin{array}{rcl}
\widehat{P}_{1\,a} &=& 2 d_a p_a\;, \hspace{3em} 1 \leq a \leq 5 \\
\widehat{P}_{2\,2} &=& 4 p_2 \\
\widehat{P}_{2\,3} &=& 0 \\
\widehat{P}_{2\,4} &=& 4 p_4 \\
\widehat{P}_{2\,5} &=& 4 p_5 \\
\widehat{P}_{3\,3} &=& 108 \left(p_1 -p_2\right)^{2} \\
\widehat{P}_{3\,4} &=& 18\,\left( - 2\,p_1\,p_2+ 2 {p_2}^2+ p_5 \right) \\
\widehat{P}_{3\,5} &=& 12\,\left( \,p_2\,p_3+ p_1 \,p_4 -p_2 \, p_4 \right) \\
\widehat{P}_{4\,4} &=& 12 \,\left({p_2}^2 + p_5 \right)\\
\widehat{P}_{4\,5} &=& 4\,\left( \,p_2\,p_3+3\, p_1 \,p_4 -p_2 \, p_4 \right) \\
\widehat{P}_{5\,5} &=& \frac{4}{3} \left(p_3 \, p_4 + {p_4}^2 + 9\, p_1 p_5 \right)\;,
\end{array}$$ where the ordered set $(d_1,d_2,d_3,d_4,d_5)=(2,2,3,3,4)$ specifies the degrees of the polynomials of the MIB.

