I am looking for real $n \times n$ matrices such that trace$(AA)=0$.
So far I have found out that
$A^2=BC-CB$ for some $n \times n$
The nullspace of trace is generated by $\{E_{ij} | i \neq j\} \cup \{H_{i,i+1} | 1 \leq i \leq n-1\} $ where $E_{ij}$ is the $n \times n$ matrix with $a_{ij}=1$ and $0$ elsewhere and $H_{ij}=E_{ii}-E_{jj}$, the dimension of the nullspace is $n^2-1$, so these matrices are the basis of the nullspace.1
However I am stuck figuring out how to relate that to $A^2$. I welcome any hints!