Let $g$ be a Lie algebra. Consider the adjoint action $g \times g \otimes g \otimes g \to g \otimes g \otimes g$ given by \begin{align} x.(a \otimes b \otimes c) = [x, a] \otimes b \otimes c + a \otimes [x, b] \otimes c + a \otimes b \otimes [x, c], \quad x, a, b, c \in g. \end{align} What are the $g$-invariants of $g \otimes g \otimes g$ under this action? In particular, let $g = sl_2$. What are the $sl_2$-invariants of $sl_2 \otimes sl_2 \otimes sl_2$ under adjoint action? Any help will be greatly appreciated!
Edit: it seems that we need to compute all $a \otimes b \otimes c \in g \otimes g \otimes g$ such that $x.(a \otimes b \otimes c) = 0$ for all $x \in g$.
This is a low-tech (brute force) approach, without revealing the answer, that is only reasonable for a small-dimensional algebra and a small number of tensor factors. It's worth doing, though, since it forces you to think about how to encode things in a meaningful way so that a computer algebra system can do the linear algebra for you. I will describe the calculation for $\mathfrak{g} = \mathfrak{sl}_2$, and you can generalize.
Choose a basis $\{e, h, f\}$ for $\mathfrak{sl}_2$ with brackets: $$ [h,e] = 2e, \qquad [h,f]=-2f, \qquad [e,f]=h. $$
Use this to construct a basis for the $3^3=27$-dimensional space $\mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g}$. Consider the action of $x=e$ on this space, which is represented by a $27 \times 27$ matrix. Compute its kernel. Repeat this for $h$ and $f$. Find the intersection of these three subspaces.