Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.

If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' fourier transform matrix:

$$F_N = \frac{1}{\sqrt{N}} \begin{bmatrix} 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\ 1&\omega^2&\omega^4&\omega^6&\cdots&\omega^{2(N-1)}\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^{3(N-1)}\\ \vdots&\vdots&\vdots&\vdots&&\vdots\\ 1&\omega^{N-1}&\omega^{2(N-1)}&\omega^{3(N-1)}&\cdots&\omega^{(N-1)(N-1)} \end{bmatrix}$$ (where $\omega=\exp(2i\pi/N)$, see the wikipedia page on Quantum Fourier Transform)

what are the possibilities for the Lie algebra generated (i.e. all linear combinations of all bracket expressions) as $\left<A, \tilde{B}\right>_{Lie}$? I.e. what algebras can be generated this way by varying $B$ while keeping it diagonal.

Specifically, will the Lie group associated to whatever algebra is generated contain a subgroup isomorphic to any non-trivial Clifford group?

realLie subalgebra generated by $A,\tilde{B}$? $\endgroup$