Mathematical objects that satisfy commuting property and $A^2=0$ Is there a set of any numbers, matrices or their generalizations that satisfies all of the following?
1) $A^2=B^2=C^2=D^2... =0$ where $A,B,C,D..$ are unequal mathematical objects
2) Objects in the set commute.
3) Other products of objects that do not involve the squared number of an object are not zero.
Also, what would be the restriction on the number of objects?

One example is the group $\mathbb{Z}_2 \times\mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$.
