= Connected nondegenerate transition chain generates special unitary control
{title2=$\operatorname{Lie}\{iH_0,iH_1\}\supseteq\mathfrak{su}(N)$}
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 $d_j\ne0$ 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.
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