Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 1 / d

Put $S=X_1+X_2$. By symmetry,Their sum is $S$, which is measurable with respect to $\mathcal{G}=\sigma (S)$, soThereforeThis also follows from the Gaussian conditional expectation formula because $\mathrm{Cov}(X_1,S)/\mathrm{Var}(S)=1/2$.