 Connected nondegenerate transition chain generates special unitary control

Let $H_0$ be diagonal and $H_1={\sum }_{j=1}^{N-1}{d}_j(E_{j,j+1}+E_{j+1,j})$, with every $d_j\ne 0$ and the adjacent absolute energy gaps distinct and nonzero. The squared drift adjoint action has distinct eigenvalues on these coupling components, so Lagrange interpolation isolates every adjacent symmetric coupling. One more commutator gives its antisymmetric partner. Their mutual brackets produce adjacent traceless diagonals, while brackets along the connected chain give all longer-range matrix units. Thus the special unitary Lie algebra is generated. Any nonzero-trace generator also supplies the central direction, giving all of $\mathfrak{u}(N)$. This is a sufficient condition in the finite-dimensional freely timed real-control model; connectivity alone does not remove frequency degeneracies.