I'm trying to understand the $\mathfrak{sl}(2,\mathbb{C})$ irreducible representation.
It is known that any irreducible representation $\pi : \mathfrak{sl}(2,\mathbb{C}) \xrightarrow{} GL(V)$, with $V$ finite dimensional vector space is isomorphic to the representation $\pi_m : \mathfrak{sl}(2,\mathbb{C}) \xrightarrow{} GL(\mathbb{V}_m)$ where $\mathbb{V}_m$ are the homogenuous polinomial on $\mathbb{C}^2$ of degree $m=\dim(V)-1$. In particular, by using Clebsch–Gordan theory for $\mathfrak{sl}(2,\mathbb{C})$ it is possible to decompose spaces as the space of symmetric and skew symmetric tensor over $\mathbb{V}_m$ into irreducibles.
In particular, the abstract irreducible decomposition of $S^2 \mathbb{V}_2$ (the space of symmetric two tensor) is $S^2 \mathbb{V}_2=\mathbb{V}_4 \oplus \mathbb{V}_0$
I'm puzled by the following sentence: "the $\mathbb{V}_0$ factor indicates that there exists a unique (up to scale) sl(2; C)-invariant symmetric bilinear form g = (·, ·) on $\mathbb{V}_2$."
More generally in $\mathbb{V}_m$ if $m$ is even we have a symmetric bilinear form, while if $m$ is odd we have a skwe symmetric bilinear form.
I know that such form is obtained by taking the higest weight element in $\mathbb{V}_0$ in the decomposition of $S^2 \mathbb{V}_m$ or $\Lambda^2 \mathbb{V}_m$ (in the basis given by standard $\mathfrak{sl}(2)$ triple) and applying to it the interwinding $\mathbb{C}^2 \rightarrow (\mathbb{C}^2)^*$, but i can't figure out why such form has to be invariant and why it is unique (up to a constant). What is the connection between $\mathbb{V}_0$ and the $\mathfrak{sl}(2)$-invariant forms?