presented in [@kn:homolie; @kn:mostow], is unnecesary in the category of algebraic groups.

For the proof of Theorem \[theo:geocarac\] we need some preparation.

\[lema:gene\] Assume that $H \subseteq G$ is a closed inclusion of affine algebraic groups, if $0 \neq I \subseteq \Bbbk[G]$ is an $H$ stable ideal, there is a non zero element $f \in I$ and an extendible character $\chi$ of $H$ such that $x\cdot f = \chi(x)f$ for all $x \in H$ and that $f(1)\neq 0$.

Take $V$ a simple rational $H$–submodule of $I$. Choose a basis $\{e_1,\cdots,e_n\}$ with the property that $e_1(1)=1,e_i(1) = 0$ for $i=2,\cdots,n$. Take $V^*$ the linear dual of $V$ that is a rational $H$–module, and apply Theorem \[theo:obse1\] to obtain an extendible character $\chi$ of $H$ and inclusion $\iota : V^* \to \Bbbk[G]_{\chi^{-1}}$ with the property that $\iota(e_1^*)(1)\neq 0$. The equivariance property of $\iota$ reads as: $\iota(x\cdot\alpha)=x\cdot_{\chi^{-1}} \iota(\alpha)=\chi^{-1}(x)x\cdot \iota(\alpha)$. It is clear that $\sum e_i \otimes e_i^*$ is $H$–stable, i.e. for all $x \in H$ we have that $x\cdot e_i \otimes x\cdot e_i^*= \sum e_i \otimes
  e_i^*$, if we apply ${\ensuremath{\mathrm{id}}}\otimes \iota$ to this equality, we deduce that $\sum e_i \otimes \iota(e_i^*)=\sum x\cdot e_i \otimes
  \iota(x\cdot e_i^*)= \chi^{-1}(x)\sum x\cdot e_i \otimes x \cdot
  