$L({{\mathscr B}})$ generates ${{\mathscr B}}$, the latter is equivalent to $L({{\mathscr B}})$ being contained in the image of ${{\mathbb R}}G$. This is what we will show.

Since $L({{\mathscr B}})$ is compact, $({\operatorname{Sym}}^2 F)^{L({{\mathscr B}})}$ has a ${L({{\mathscr B}})}$-invariant supplement $({\operatorname{Sym}}^2F)^{{L({{\mathscr B}})}\not=1}$ in ${\operatorname{Sym}}^2 F$. This is also $G$-invariant and so $$({\operatorname{Sym}}^2F)^{G,{L({{\mathscr B}})}\not=1}:=({\operatorname{Sym}}^2F)^G\cap  ({\operatorname{Sym}}^2F)^{{L({{\mathscr B}})}\not=1}$$ is a $G$-invariant supplement of $({\operatorname{Sym}}^2 F)^{L({{\mathscr B}})}$ in $({\operatorname{Sym}}^2 F)^G$. In view of our hypotheses and Lemma \[lemma:quadratic\], Theorem \[endo\] will follow from:

\[lemma:endo\] We have $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}=0$.

The proof of Lemma \[lemma:endo\] involves a Hilbert scheme. We write ${{\mathbb P}}$ for the projective space $\check{{\mathbb P}}(F)$ of hyperplanes of $F$. The very general $G$-curve yields a canonical embedding $C\subset{{\mathbb P}}$. Let us denote by ${{\mathscr H}}$ the open subscheme of the irreducible component of ${\operatorname{Hilb}}({{\mathbb P}})$ which parameterizes canonically embedded smooth projective curves. The $G$-equivariant canonical embeddings of $G$-curves in ${{\mathbb P}}$ of our given topological type define an irreducible closed subscheme ${{\mathscr H}}^G$ of ${{\mathscr H}}$. The action of $G$ on $F$ induces one on ${{\mathscr H}}$ and then ${{\mathscr H}}^G$ is an irreducible component of the fixed point locus for this action.

\[quadric\]Let $X$ be a $G$-invariant quadric hypersurface of 