Listing all nilpotent matrices of $\mathcal{A} = \{c_0I+c_1E_1,c_2E_2\mid c_i\in K\}$

Question

How to solve the following problem:

Let $\mathcal{A} = \{c_0I+c_1E_1+c_2E_2\mid c_i\in K\}$, where $I,E_1,E_2\in \mathcal{M}_n(K)$ and $E_1,E_2$ are idempotent orthogonal matrices, that is $E_1^2 = E_1, E_2^2 = E_2$ and $E_1E_2=E_2E_1=O$. $K$ is a field. We also assume that $E_1,E_2\neq O$ and $E_1+E_2\neq I$. The set $\{I,E_1,E_2\}$ forms a basis for subalgebra $\mathcal{A}$.

List all nilpotent matrices of $\mathcal{A}$, that is matrices $A\in\mathcal{A}$, such that $A^m=0$ for some $m>0$.

I have tried solving this, but I'm not sure of my solution and want to verify my approach.

Answer 1

Hint: Let $$A=c_0I+c_1E_1+c_2E_2$$

Then $$A E_1 =(c_0+c_1)E_1$$ $$A E_2 =(c_0+c_2)E_2$$

This shows that $c_0+c_1$ and $c_0+c_2$ are eigenvalues of $A$ with any non-zero column of $E_1$ respectively $E_2$ as eigenvectors.

Now, if $A$ is nilpotent, all its eigenvalues are ....

Hint 2 $$(I-E_1-E_2)^2=??$$