Why the action of $\mathfrak{g}_2$ on the standard representation preserves a quadratic form?

Question

In the representation theory book by Fulton and Harris, it is proven that Sym$^2V\cong \Gamma_{2,0}\oplus \mathbb{C}$ where $V$ is the standard representation of $\mathfrak{g}_2$. I did not read the proof in detail. My question is how does this isomorphism imply that the action of $\mathfrak{g}_2$ on $V$ preserves a quadratic form?

Answer 1

We get from that isomorphism a nonzero morphism of representations $\mathrm{Sym}^2(V)\to \Bbb C$. We know that morphisms $V\otimes V\to \Bbb C$ correspond to bilinear forms on $V$ preserved by $\mathfrak g_2$. The corresponding bilinear form is symmetric if and only if the map factors over $\mathrm{Sym}^2(V)$, by the defining relations of $\mathrm{Sym}^2$. Thus a nonzero morphism $\mathrm{Sym}^2(V) \to \Bbb C$ corresponds to a symmetric bilinear form on $V$ preserved by $\mathfrak g_2$ or equivalently, a preserved quadratic form.