of a minimal set of explicit algebraic relations providing a cylindrical decomposition [@Coste] for the semi-algebraic subset $\widehat S^{(4)}$ of $\real^{5}$ would be a really hard task (for the more peripheral strata, instead, the problem is much easier to solve). An immediate application of the results proved in the previous section, on the contrary, leads to a simple rational global parametrization of each stratum, as shown below.

### Stratum $\widehat S^{(4)}$.

A “typical point" of the stratum is $x_{\rm
t}=(1,1,0,0,0,0,1,1)$. The corresponding isotropy subgroup $H$ of $G$ is the $\integer_2$ group generated by the reflection representing the element diag$(-1,1,1)\in$ O(3) in $\real^8$: $$\mbox{\rm diag}(1,1,-1,-1,1,-1,1,1)\in G.$$

The vector space $V$ formed by the $H$-invariant vectors of $\real^8$ turns out to be five–dimensional: $$V = \left\{x\in \real^8 \;|\;x_3=x_4=x_6=0\right\}.$$

The elements of O(3), corresponding to elements of $\mbox{\rm
Stab }(H,G)$, are block-diagonal matrices of the form diag$(\pm
1,O)$, with $O\in $ O(2). Therefore, $\mbox{\rm Stab }(H,G)$ is isomorphic to a group $\integer_2 \times$O(2) and the quotient group $K=\mbox{\rm Stab }(H,G)/H$ is the representation in $V$ (induced by the representation $G$ of O(3)) of the O$_1(2)$ subgroup of O(3), formed by the rotations around the first axis.

Using coordinates $v=(x_1, x_2,x_5,x_7,x_8)$ for a vector 