I am wondering about how to characterise the more general traceless matrix over $\Bbb C$
$$ \operatorname{Tr} (T) = 0$$
and having unit Hilbert-Schmidt norm,
$$ \| T \|^2 = \operatorname{Tr} \left( T T^* \right) = 1$$
For a normal matrix with real eigenvalues, it is very intuitive that the eigenvalues are the cosines of a force in a regular poligon,
$$ k_n = \cos ( 2 \pi n / d + \delta)$$ with $d$ the dimension of the matrix. Perhaps in any case I should add the condition that no subset of eigenvalues is traceless, or some other simplicity condition, but that seems to be all. I am no very worried about it because most of my cases of use are 3d matrices.
Now, for complex eigenvalues I expected that it was just multiply times a global phase $e^{i \pi \phi}$. But then I noticed that I can add a whole set of imaginary eigenvalues $$w_n = i \cos ( 2 \pi n / d + \theta)$$ with a different phase, so that $$k_n \cos \phi + w_n \sin \phi$$ still add to zero and total norm one. I see this is because
$$\|(T+iQ)\|^2 = \|T\|^2 + \|Q\|^2 + i \operatorname{Tr} \left( Q T^* - T Q^* \right)$$
so if $T$ is self adjoint I can add in this way any other self adjoint operator commuting with it.
So now I am a bit lost about which is the more general form for the eigenvalues if they exist, and of course for the whole matrix. I guess this is a very well known problem, surely I am only lacking vocabulary to search about.
In summary, what I expect is to have some clear characterisation of the matrices and of their eigenvalues. It could be possible to have some general formula for the diagonal case, and then extend via unitary transformation, or proceed via some definition of "simple traceless matrix" so that a general traceless would be a weighted sum of "simple" matrices.