Suppose $X = \mathbb{Q}-span \langle x_1, x_2, x_3 \rangle \cong \mathbb{Q}^3$ is a $3$-dimensional $\mathbb{Q}$-vector space with some basis $(x_1, x_2,x_3)$. We let the symmetric group $S_4$ act on $X$ as the product of the standard representation and the sign representation (so $X$ is an irrep of $S_4$). Consider the symmetric product $S^+(X)$ of $X$, where the $^+$ indicates that we're only considering non-zero degrees (i.e. we disregard the $0$-degree factor $\mathbb{Q}$ in the symmetric product).
Question: How can I describe the space of invariants $(S^+(X))^{S_4}$?
Some ideas: I know that we can identify the (non-zero degree) symmetric product of $X$ with the space of positive degree polynomials in $3$ variables over $\mathbb{Q}$: $$S^+(X) \cong \mathbb{Q}_{>0}[x_1, x_2, x_3].$$ So what I want to find are really the invariant polynomials $( \mathbb{Q}_{>0}[x_1, x_2, x_3])^{S_4}$.
More information about the representation I'm considering and my specific choice of basis: The elements $x_1, x_2, x_3$ I've chosen are such that the generating transpositions $(1, 2), (2, 3), (3,4) \in S_4$ have the following effect (according to my calculations): \begin{align*} (1,2): \hspace{2mm} &x_1 \mapsto -x_1, \hspace{3mm} x_2 \mapsto -x_1+x_2+x_3, \hspace{3mm} x_3 \mapsto -x_3 \\ (2,3): \hspace{2mm} &x_1 \mapsto -x_1, \hspace{3mm} x_2 \mapsto -x_3, \hspace{3mm} x_3 \mapsto -x_2 \\ (3,4): \hspace{2mm} &x_1 \mapsto x_3, \hspace{3mm} x_2 \mapsto -x_2, \hspace{3mm} x_3 \mapsto x_1 \end{align*}
I've calculated that for instance the polynomial $$g = (-x_1 + x_2 + x_3)x_1x_2x_3 \in \mathbb{Q}_{>0}[x_1, x_2, x_3]$$ is invariant under the action of the $(1, 2), (2, 3), (3,4) \in S_4$. Since these transpositions generate $S_4$, it follows that $g \in ( \mathbb{Q}_{>0}[x_1, x_2, x_3])^{S_4}$. But does the basis I've chosen matter in any way? And how do I find the entire space of invariant polynomials?
Edit: I would also greatly appreciate a reference where ways of finding invariant polynomials (in simple cases) are treated. I have the same problem also arising for actions of some semidirect and direct products involving $S_4$ and $S_2$, instead of just $S_4$.
Maybe it's also possible to treat this using computer programming. Unfortunately, I have zero experience with computer algebra programs. If this is the way to here though, then I'd be happy to learn about it.
I'm glad for any help!