$E[X_2 | X_1] $ on Gaussian (as the best predictor)

Question

$X_1, X_2 $ are jointly Gaussian.

For Gaussian, $E[X_2 | X_1] = E^L[X_2 | X_1]$ with mean zero $X_2, X_1$, where $E^L[Y|X=x] = W^*x - b^*$ and $W^*, b^* =\text{argmin}_{W,b} |Y-WX-b|$.

How do that show it is the case for $E[X_2 | X_1] = E^L[X_2 | X_1]$?