TITLE: To describe an invariant trivector in dimension 8 geometrically
QUESTION [11 upvotes]: $\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$.
For an integer $p\ge 0$, write $R_p=S^p R$; then $R_1=R$ and  $\dim R_p=p+1$.
Using Table 5 in the book of Onishchik and Vinberg, I computed that the representation
$$ R_2\otimes\Alt^2 R_4 $$
contains the trivial representation with multiplicity one.
I used the table as a black box.

Question. Let $V\subset  R_2\otimes\Alt^2 R_4$ denote the corresponding one-dimensional subspace.
How can one describe $V$ as a subspace geometrically?

Motivation: I want to consider a $\operatorname{PGL}(2,k)$-fixed trivector
$$v\in V\subset R_2\otimes\Alt^2 R_4\subset \Alt^3(R_2\oplus R_4)$$
of  the 8-dimensional vector space $W=R_2\oplus R_4$
over a field $k$ of characteristic 0,
and then to twist all this using a Galois-cocycle of $\operatorname{PGL}(2,k)$. For this end I need a geometric description of $V$.
Feel free to add/edit tags!

REPLY [12 votes]: Here's another very nice (but still algebraic) interpretation that explains some of the geometry:  Recall that $\operatorname{SL}(2,\mathbb{C})$ has a $2$-to-$1$ representation into $\operatorname{SL}(3,\mathbb{C})$ so that the Lie algebra splits as
$$
{\frak{sl}}(3,\mathbb{C}) = {\frak{sl}}(2,\mathbb{C})\oplus {\frak{m}}
$$
where ${\frak{m}}$ is the ($5$-dimensional) orthogonal complement of ${\frak{sl}}(2,\mathbb{C})$ using the Killing form of ${\frak{sl}}(3,\mathbb{C})$.   Note that ${\frak{m}}$ is an irreducible ${\frak{sl}}(2,\mathbb{C})$-module, and that every element $x\in {\frak{sl}}(3,\mathbb{C})$ can be written uniquely as $x = x_0 + x_1$ with $x_0\in {\frak{sl}}(2,\mathbb{C})$ and $x_1\in{\frak{m}}$.  Note also that $[{\frak{m}},{\frak{m}}]= {\frak{sl}}(2,\mathbb{C})$.
This defines the desired pairing ${\frak{sl}}(2,\mathbb{C})\times \bigwedge\nolimits^2({\frak{m}})\to\mathbb{C}$:  Send $(x_0,y_1,z_1)$ to $\operatorname{tr}(x_0[y_1,z_1])$.  Of course, this makes the $\operatorname{SL}(2,\mathbb{C})$-invariance of the pairing obvious.