= Solution
Assume $c_1c_2\ne0$, as required for a connected chain and the supplied divisions. The <single-excitation subspace> has dimension three, so its <Dynamical Lie algebra> $\mathfrak g_1=\operatorname{Lie}_{\mathbb R}\{iA,iB\}$ is contained in $\mathfrak u(3)$. The intended conclusion from nine independent skew-Hermitian directions is $\boxed{\mathfrak g_1=\mathfrak u(3)}$, giving <unitary operator controllability>, <density operator controllability> and <pure-state controllability> on this sector. Here is a direct generation argument that does not assume an independent identity control or rely on the normalization of the printed intermediate matrices.
Here $c_n$ denotes the hopping entry of the supplied reduced $A$. With the stated <Pauli matrices>, restricting $c_n(X_nX_{n+1}+Y_nY_{n+1})$ instead gives hopping $2c_n$; absorbing that common factor into the reduced coupling does not change the <Dynamical Lie algebra>.
Write $R_{mn}=E_{mn}-E_{nm}$ and $Y_{mn}=i(E_{mn}+E_{nm})$ using <matrix units>. The physical generators give
$$
R_{12}=\frac{[iB,iA]}{2c_1},\qquad Y_{12}=-\frac12[iB,R_{12}],\qquad D_{12}=\frac12[R_{12},Y_{12}]=i(E_{11}-E_{22}).
$$
Next isolate the second transition and generate its other direction:
$$
Y_{23}=\frac{iA-c_1Y_{12}}{c_2},\qquad R_{23}=[D_{12},Y_{23}],\qquad D_{23}=\frac12[R_{23},Y_{23}]=i(E_{22}-E_{33}).
$$
Finally $R_{13}=[R_{12},R_{23}]$ and $Y_{13}=[R_{12},Y_{23}]$. The three $R_{mn}$, three $Y_{mn}$ and two diagonal differences are eight independent traceless skew-Hermitian matrices, hence span $\mathfrak{su}(3)$. Because $\operatorname{Tr}(iB)=i\ne0$, subtracting the traceless part of $iB$ also supplies $iI_3$, completing $\mathfrak u(3)$. This is the trace-direction step in <identity augmentation in quantum controllability> and the three-site instance of <Endpoint control of an XX spin chain>. If a coupling vanishes, the chain disconnects and the full-sector conclusion fails.
There is a redundant direction in the printed final pair of <commutators>: with the printed intermediate definitions, both are proportional to $E_{13}-E_{31}$. Thus that pair cannot itself establish the stated independence. The mixed <commutator> $[R_{12},Y_{23}]=i(E_{13}+E_{31})$ above provides the missing direction and establishes the required conclusion independently.
Let $\mathfrak g_{\mathrm{full}}$ be the algebra on the full eight-dimensional space. Restriction to the invariant single-excitation sector is a surjective <Lie algebra homomorphism> $R:\mathfrak g_{\mathrm{full}}\to\mathfrak u(3)$. If $\dim\mathfrak g_{\mathrm{full}}=9$, the <rank-nullity theorem> gives $\ker R=0$, so
$$
\boxed{\mathfrak g_{\mathrm{full}}\cong\mathfrak u(3)\cong\mathfrak{su}(3)\oplus\mathfrak u(1)}.
$$
This is an abstract algebra isomorphism; on the full <Hilbert space> it is a reducible eight-dimensional representation preserving sectors of dimensions $1,3,3,1$. The sector actions are linked by the same nine generators, rather than independently controllable blocks. In particular, this is not $\mathfrak u(8)$ or $\mathfrak{su}(8)$, so it does not imply full-system controllability. The full spin Hamiltonians are traceless, so their algebra is embedded in $\mathfrak{su}(8)$ even though its abstract type is $\mathfrak u(3)$; its central direction is not an unrestricted overall phase on the full space.
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