The determinant of the matrix $\widehat P(p)$ factorizes and only one of the two real irreducible factors, that we shall call $A(p)$, turns out to be [*active*]{} [@682], that is, to be relevant in the determination of the boundary of the orbit space $\real^8/G$:

$$\label{attivo}
\begin{array}{rcl}
A(p)&=& 3\,{{p_2}}^3\,{{p_3}}^2 + 18\,{p_1}\,{{p_2}}^2\,{p_3}\,{p_4} -
  18\,{{p_2}}^3\,{p_3}\,{p_4} +
  27\,{{p_1}}^2\,{p_2}\,{{p_4}}^2 \\
  && -
  54\,{p_1}\,{{p_2}}^2\,{{p_4}}^2 + 27\,{{p_2}}^3\,{{p_4}}^2 +
  {p_3}\,{{p_4}}^3 - 9\,{p_2}\,{p_3}\,{p_4}\,{p_5}  \\
  &&
  -9\,{p_1}\,{{p_4}}^2\,{p_5} + 9\,{p_2}\,{{p_4}}^2\,{p_5} -
  27\,{p_1}\,{p_2}\,{{p_5}}^2 + 27\,{{p_2}}^2\,{{p_5}}^2 +
  9\,{{p_5}}^3\;.\\
\end{array}$$

The relations defining strata of dimension $<4$ in the orbit space of (O(3),$\real^8$) are summarized in . The isotropy subgroup lattice, with the possible phase transitions between bordering strata is shown in .

Parametrization of the strata
-----------------------------

The relations assuring that $\widehat P(p)\ge 0$ and has rank 4, define a unique four–dimensional stratum $\widehat S^{(4)}$ in the orbit space. Using well known matrix theory results, these conditions could be explicitly written, for instance, in the form $A(p)=0$ and $M_i(p)>0$, $i=1,\dots ,4$, where $M_i$ is the sum of the principal minors of order $i$ of the matrix $\widehat P(p)$: a cumbersome set of conditions that it is not worthwhile to write down explicitly.

Even in this simple example one immediately realizes that the choice 