Write corresponding eigenvectors for eigenvalues of a linear transformation

Question

Let $V$ be a finite-dimensional inner product space, and assume given a self-adjoint linear transformation $T:V\to V$ such that $T^3 = T$.

How do I show that the only possible eigenvalues of $T$ are $\lambda = 0, 1,$ or $−1$? And how do I write the corresponding eigenspaces?

That is what I did so far:

We have $T^3 = T$.

So if $\lambda$ is an eigenvalue of $T$ then we have $\lambda^3 = \lambda$.

$\lambda^3 = \lambda$

$\lambda^3-\lambda=0$

$\lambda(\lambda^2 - 1) = 0$

so $λ = 0$ or $λ =1$ 0r $λ = -1$.

Now, how do we write the eigenvectors?