called the [*variety $\cal Z$ of the relations*]{} (among the elements of the MIB). If $G$ is coregular, ${\cal Z}=\real^q$.

Since $G$ is a compact group, the orbits of $G$ are separated by the elements of a MIB of $G$ (for an elementary proof see, for instance, [@020a]), [*i.e.*]{}, at least one element of a MIB of $G$ takes on different values at two distinct orbits. Thus, the elements of a MIB of $G$ yield a parametrization of the points of $\real^{n}/G$, that turns out to be also smooth, since the [*orbit map*]{} $p:\, \real^{n} \longrightarrow \real^{q}$, which maps all the points of $\real^{n}$ lying on an orbit of $G$ onto a single point of $\real^{q}$, induces a diffeomorphism of $\real^{n}/G$ onto a semialgebraic $q$–dimensional connected closed subset of $\real^{q}$. Like all semialgebraic varieties [@840], $p(\real^n)$ presents a natural stratification in connected semialgebraic sub-varieties, called [*primary strata*]{}, which turn out to be the connected components of the isotropy-type strata[^1].

A characterization of the image $p(\real^n)$ of the orbit space of $G$ as a semi-algebraic variety and of its primary strata can be easily obtained through a matrix $\widehat{P}(p)$, defined only in terms of the $G$-invariant Euclidean scalar products between the 