Let $\mathcal{S_3}$ be the collection of $3\times 3$ skew-symmetric matrices with integer entries. There is no non-zero $D\in \mathcal{S_3}$ that satisfies the following equation $$[D,[D,[D,N]]]=0$$ for all $N\in \mathcal{S_3}$ where $[\cdot,\cdot]$ is the usual commutator of matrices ($[A,B]=AB-BA$).
How can I prove or disprove this for all $n$, i.e., show or disprove that there is no $D\in \mathcal{S_n}\setminus {0}$ such that $[D,[D,[D,N]]]=0$ for any $N\in \mathcal{S_n}$?
I've attempted to prove it by considering the generators of $\mathcal{S_n}$ and then computing its commutator with one another. I then expressed $D$ as the linear combination of those generators. It is very tedious (even if I let $N$ to be one of the generators) and I did not get any result. I am now looking for other ways to prove or disprove it. Does $\mathcal{S_n}$ have properties you that might be useful in this problem?