\iota(e_i^*)$. Then, the element $f = \sum e_i\iota(e_i^*) \in I$ satisfies the following equivariance property $f=\sum e_i
  \iota(e_i^*)=\chi^{-1}(x)\sum (x\cdot e_i) (x \cdot
  \iota(e_i^*))=\chi^{-1}(x)x\cdot \sum e_i \iota(e_i^*)=\chi^{-1}(x)
  x \cdot f$. Moreover, $f \neq 0$ as $f(1)=\sum e_i(1)
  \iota(e_i^*)(1) = e_1(1)\iota(e_1^*)(1)=\iota(e_1^*)(1)\neq 0$.

The following characterization of observability seems to have appeared for the first time in [@kn:nosotros2 Chapter 11, Section 5].

[[@kn:nosotros2 Theorem 11.5.1]]{} \[theo:charcinv\] Assume that $H \subseteq G$ is a closed inclusion of affine algebraic groups. Then $H$ is observable in $G$ if and only if, for every $H$–stable ideal $I \subseteq \Bbbk[G]$, there is a non zero element $f \in I$ such that $x \cdot f =f$ for all $x \in H$. Also, $H \subseteq G$ is observable if and only if for every closed proper subset of the quotient space $C \subseteq G/H$ there is a non zero invariant polynomial such that $f(C)=0$.

In accordance with the lemma just proved, we can find $h \in I$ with the property that $x\cdot h = \chi(x)h$ for some extendible character $\chi$ and with $h(1)=1$. If $H$ is observable, the character $\chi^{-1}$ is also extendible and then there is an element $g \in \Bbbk[G]$ such that $g(1)\neq 0$ and $x \cdot 