Lie-algebraic (g-sim) loss reconstruction

Let $\mathfrak g=\langle\{i A_l\}\rangle_{\operatorname{Lie}}\subseteq\mathfrak u(2^n)$ be the dynamical Lie algebra generated by the skew-Hermitian generators $i A_l$ of a variational circuit (with $A_l$ Hermitian), and let $\{B_j\}_{j=1}^{\dim\mathfrak g}$ be a Hilbert–Schmidt-orthonormal basis of the Hermitian sector $i\mathfrak g$, so that $\{i B_j\}$ is a basis of $\mathfrak g$ and the quantum data $\operatorname{Tr}[\rho B_j]$ are real. For every circuit $U=\prod_k e^{i A_k}$ with Hermitian generators $A_k\in i\mathfrak g$ and every Hermitian observable $O\in i\mathfrak g$, the loss reconstructs exactly as $$\operatorname{Tr}\!\big[\,U\rho\,U^{\dagger}O\,\big]=\sum_{j=1}^{\dim\mathfrak g}\big\langle B_j,\,\mathcal H(O)\big\rangle_{\mathrm{HS}}\;\operatorname{Tr}[\rho B_j],$$ where $\mathcal H(O)=U^{\dagger}OU$ is the Heisenberg-evolved observable and $\langle B_j,\mathcal H(O)\rangle_{\mathrm{HS}}=\operatorname{Tr}[B_j\,\mathcal H(O)]$ are its basis coordinates.