Dynamical Lie algebra of a local single-qubit-gate circuit

For an $n$-qubit variational circuit built from single-qubit gates, the dynamical Lie algebra is spanned by the $3n$ single-site Pauli operators $\{X_j,\,Y_j,\,Z_j\}_{j=1}^{n}$ and equals the orthogonal direct sum of $n$ mutually commuting single-qubit copies of $\mathfrak{su}(2)$: $$\mathfrak g \;=\; \bigoplus_{j=1}^{n}\mathfrak{su}(2)_j,\qquad \dim\mathfrak g = 3n.$$