= Endpoint control of an XX spin chain
{c}
{title2=$\mathfrak g|_{\mathcal H_1}=\mathfrak u(N)\ (N\ge3)$}
In the <single-excitation subspace>, take a connected real nearest-neighbor hopping matrix $A$ with every coupling nonzero and endpoint control $B=\operatorname{diag}(-1,1,\ldots,1)$. 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 $N\ge3$, $\operatorname{Tr}B=N-2\ne0$ 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.
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