Suppose that there is a set of matrices.
1) $A^2 =0, C^2=0, E^2=0 .....$ and $AB+BA=0$, $CD+DC=0$... where $A,B,C..$ are matrices.
2) Matrices in the set either anticommute or commute.
3) $AD+BC \neq 0,... $
Is there any set of matrices that satisfies all of the aforementioned?
Edit: all matrices are not equal to each other.