g-sim: the evolved observable stays in the algebra

For a variational circuit $U=\prod_k e^{i A_k}$ whose skew-Hermitian generators $i A_k$ lie in the dynamical Lie algebra $\mathfrak g$, the Heisenberg-evolved image of any Hermitian observable $O\in i\mathfrak g$ remains in the algebra: $$\mathcal H(O)=U^{\dagger}OU\in i\mathfrak g.$$ Equivalently, the adjoint action of every gate preserves $\mathfrak g$, $e^{-S}\,\mathfrak g\,e^{S}=\mathfrak g$ for $S\in\mathfrak g$, so the $\dim\mathfrak g$-dimensional algebra is closed under Heisenberg evolution.