I want to make a set of matrices that satisfies all the following:
1) $A^2 = B^2$, $C^2 =D^2$..... where $A,B,C,D...$ are matrices
2) $AB = 0$, $CD = 0$.....
3) All matrices in the set commute.
4) $AC \neq 0$, $AD \neq 0$, $BC \neq 0$, $BD \neq 0$.... so other than $AB$, $CD$ ... multiplication is always nonzero.
Also, is there any mathematical object that satisfies all of these?