$V_\chi\oplus V_{\chi*}$) and that the obvious ${{\mathbb R}}$-algebra homomorphism $$\label{eqn:rdec}
\textstyle {{\mathbb R}}G\to  \big(\prod_{\chi\in X({{\mathbb C}}G)_+} {\operatorname{End}}_{{\mathbb R}}(V_\chi({{\mathbb R}})) \big)  \times \big(\prod_{\chi\in X({{\mathbb C}}G)_-} {\operatorname{End}}_{{\mathbb K}}(V_\chi)\big)  
\times \big(\prod_{\chi\in X({{\mathbb C}}G)_o} {\operatorname{End}}(V_\chi)\big).$$ is an isomorphism (see for instance [@Serre], §13.2).

For $\chi\in X({{\mathbb C}}G)$ we put $F[\chi]:={\operatorname{Hom}}_{{{\mathbb C}}G}(V_\chi, F)$, so that the obvious evaluation map $\oplus_{\chi\in X({{\mathbb C}}G)} V_\chi\otimes F[\chi]\to F$ is an isomorphism of $G$-modules. Via this isomorphism, the image of ${{\mathbb C}}G$ in ${\operatorname{End}}(F)$ is identified with the product of the ${\operatorname{End}}(V_\chi)$ taken over all $\chi\in X({{\mathbb C}}G)$ with $F[\chi]\not=0$. The generators $\alpha_\chi$ determine an isomorphism $$({\operatorname{Sym}}^2 F)^G
\cong \big(\underset{\chi\in X({{\mathbb C}}G)_+}\oplus {\operatorname{Sym}}^2 F[\chi]\big)  \oplus \big(\underset{\chi\in X({{\mathbb C}}G)_-}\oplus  \wedge^2 F[\chi]\big)  
\oplus \big(\underset{\chi\in X({{\mathbb C}}G)_o} \oplus F[\chi]\otimes F[\chi^*]\big).$$ We are given that this space is also equal to $({\operatorname{Sym}}^2 F)^L$. We claim that for every $\chi\in X({{\mathbb C}}G)$ and every line $\ell \subset F[\chi]$, $L$ preserves $V_\chi\otimes\ell$. To see this, note that with any $q\in {\operatorname{Sym}}^2 F$ is associated a subspace $F_q\subset F$, namely the smallest subspace of $F$ such that $q\in {\operatorname{Sym}}^2F_q$ (so that $q$ is nondegenerate, when regarded as a quadratic form on the dual of $F_q$). It is clear that any linear transformation of $F$ which fixes 