Dynamical Lie algebra of a variational circuit

A variational quantum circuit on $n$ qubits is built from Hermitian generators $\{G_l\}_{l=1}^{L}$ as the product $U(\theta) = \prod_{l=1}^{L} e^{-i \theta_l G_l}$. Its dynamical Lie algebra is the smallest real Lie subalgebra of $\mathfrak{u}(2^n)$ that contains the skew-Hermitian generators $\{i G_l\}_{l=1}^{L}$ and is closed under real-linear combinations and the matrix commutator $[A,B] = AB - BA$: $$\mathfrak{g} = \langle \{ i G_l \}_{l=1}^{L} \rangle_{\operatorname{Lie}}.$$ The formalization represents $\mathfrak{g}$ by the complex Lie subalgebra it generates (its complexification inside the general linear algebra), which has the same dimension as the real algebra.