points" in strata, from which the orbit type of the stratum can be identified [@Li; @K].

The real orthogonal linear group $G$, just defined, is coregular (see [@711]) and its isotropy subgroup at a generic point of $\real^8$ is trivial. As a consequence, the principal orbits are three–dimensional manifolds, the principal stratum has dimensions five and there are five independent elements in a MIB (see, for instance [@683]), which can be chosen in the following way:

$$\begin{aligned}
p_1 &=&\Tr\, Q^{2} + \bi{P}\cdot
\bi{P} \nonumber\\
    &=& \sum_{j=1}^{8} {x_j}^{2}\,,\nonumber\\
&&\nonumber\\
p_2 &=& \bi{P}\cdot \bi{P} \nonumber\\
    &=&  \sum_{j=6}^{8} {x_j}^{2}\,,\nonumber\\
&&\nonumber\\
p_3 &=& 6 \sqrt{2} \,\,\Tr\, Q^{3}\nonumber\\
    &=& -2\, {\sqrt{3}}\, {{x_1}}^3 + 6\, {\sqrt{3}}\, {x_1}\, {{x_2}}^2
    - 3\, {\sqrt{3}}\, {x_1}\, {{x_3}}^2 - 9\,  {x_2}\, {{x_3}}^2
      \label{BaseInt}\\
  && - 3\, {\sqrt{3}}\, {x_1}\, {{x_4}}^2+ 9\, {x_2}\, {{x_4}}^2+18\,
{x_3}\, {x_4}\, {x_5} +   6\, {\sqrt{3}}\, {x_1}\, {{x_5}}^2 \,,\nonumber\\
&& \nonumber\\
p_4 &=& 3 \sqrt{2} \,\sum_{\alpha\,\beta}P_\alpha Q_{\alpha\,\beta}
P_{\beta}\nonumber\\
 &=& -2\,{\sqrt{3}}\, {x_1}\, {{x_6}}^2  +
  6\, {x_3}\, {x_6}\, {x_7} + {\sqrt{3}}\, {x_1}\, {{x_7}}^2 -
  3\, {x_2}\, {{x_7}}^2  \nonumber\\
  && + 6\, {x_4}\, {x_6}\, {x_8}+ 6\, {x_5}\, {x_7}\, {x_8}+ \sqrt{3}\,
{x_1}\, {{x_8}}^2+3\, {x_2}\, {{x_8}}^2\,,\nonumber\\
&&\nonumber\\
p_5 &=& 6 \,\sum_{\alpha\,\beta}P_\alpha {Q^{2}}_{\alpha\,\beta}
P_{\beta}\nonumber\\
    &=& 4\,{{x_1}}^2\, {{x_6}}^2+ 3\, {{x_3}}^2\, {{x_6}}^2 +
  3\, {{x_4}}^2\, {{x_6}}^2-2\, {\sqrt{3}}\, 