I hate and am not used to finding orthogonal basis... i'll only be able to show you the eigenspaces.
An eigenvalue verifies :
Let's do it for i.
If we calculate the product on the right side, we'll have a system of equations :
From (1), if we divide all the stuff by i, we have
So the only condition is that
A vector generating the eigenspace associated to i is
Now, this will be exactly the same for -i.
If i'm not wrong, you should get :