Is $A^3=A$ a condition for idempotency of matrices?

Question

Given that $A$ and $B$ are two idempotent (square) matrices of same order, $AB+BA=AB-BA=O$ (Where $O$ is the null matrix of the same order). Prove that both $A+B$ and $A-B$ are idempotent.

I proceeded the following way:

$(A+B)^2=A^2+AB+BA+B^2$.

As $A$ and $B$ are idempotent, and $AB+BA=O$, $A^2+B^2=A+B$.

Thus, $A+B$ is idempotent, QED.

Now, for the other part, (to prove $A-B$ is idempotent), I proceed like this:

$(A+B)(A-B)=A^2-AB+BA-B^2$.

Again, as $A$ and $B$ are idempotent, and $AB-BA=O$, $(A+B)(A-B)=A-B$.

Next, $(A-B)^3=(A-B)(A-B)^2$.

$(A-B)^2=A^2-AB-BA+B^2=A+B-(AB+BA)=A+B$.

Hence, $(A-B)^3=(A-B)(A+B)=A-B$.

Thus, I prove $(A-B)^3=A-B$.

Now, I have 2 questions:

1. Does this imply that $A-B$ is idempotent or not?

2. If not, how do I proceed further to prove it to be idempotent?