Endpoint control of an XX spin chain ($\mathfrak{g}{|}_{{\mathcal{H}}_1}=\mathfrak{u}(N)(N\ge 3)$)

In the single-excitation subspace, take a connected real nearest-neighbor hopping matrix $A$ with every coupling nonzero and endpoint control $B=\mathrm{diag}(-1,1,\dots ,1)$. Endpoint commutators isolate the first transition, then propagate along the chain to generate both off-diagonal skew-Hermitian directions and adjacent diagonal differences. These generate $\mathfrak{su}(N)$. For $N\ge 3$, $\mathrm{Tr}B=N-2\ne 0$ supplies the central direction, giving $\mathfrak{u}(N)$. For $N=2$ the algebra is $\mathfrak{su}(2)$, which still gives full state controllability but omits an arbitrary global phase. A broken coupling separates an unreachable chain component.