Dynamical Lie algebra of a matchgate (free-fermion) circuit

For an $n$-qubit matchgate (free-fermion) circuit, generated by the Majorana quadratics (the $n(2n-1)$ skew-Hermitian Pauli strings quadratic in the Jordan-Wigner Majorana operators), realized for example by the transverse-field Ising chain (on-site transverse fields with nearest-neighbor Ising couplings), the dynamical Lie algebra is the free-fermion special orthogonal algebra $$\mathfrak g=\mathfrak{so}(2n),\qquad \dim\mathfrak g=n(2n-1),$$ which grows only polynomially in the qubit count $n$.