= Two adjacent transitions generate the qutrit unitary Lie algebra
{title2=$\operatorname{Lie}_{\mathbb R}\{-iH_0,-iH_1\}=\mathfrak u(3)$}
Let $H_0=\operatorname{diag}(-\omega_1,0,-\omega_2)$ and $H_1=d_1(E_{12}+E_{21})+d_2(E_{23}+E_{32})$, with nonzero $d_j$, distinct positive $\omega_j$. The squared adjoint action of $-iH_0$ has different <eigenvalues> $-\omega_1^2,-\omega_2^2$ on the two adjacent skew-Hermitian couplings. Linear combinations of $-iH_1$ and its double <commutator> with $-iH_0$ isolate both couplings. Further <commutators> give their real antisymmetric partners, two traceless diagonal generators, and both $1$–$3$ generators, hence the <special unitary Lie algebra> $\mathfrak{su}(3)$. The drift has nonzero <trace>; subtracting its traceless part supplies $iI$, giving the <Lie algebra of the unitary group> $\mathfrak u(3)$ and <unitary operator controllability> under the usual freely timed real-control assumptions.
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