Suppose I have two random variables $X_1$ and $X_2$ that are not independent of each other. Consider the linear projections of $X_1X_2$ and $X_1$ onto each other: $L(X_1X_2|X_1)$ and $L(X_1|X_1X_2)$, where $L(A|B)\equiv (E(A^2))^{-1}E(AB)B$.
Define the projection errors as: $e_1\equiv X_1X_2-L(X_1X_2|X_1)=X_1X_2-(E(X_1^2))^{-1}E(X_1^2X_2)X_1$ and $e_{12}\equiv X_1-L(X_1|X_1X_2)=X_1-(E(X_1^2 X_2^2))^{-1}E(X_1^2 X_2)X_1X_2$.
My question is, do $e_1$ and $e_{12}$ together satisfy any equality restriction or identity? (Since they are the projection erros that result from linear projections of two random variables onto each other, I guess they should be related somehow) If there isn't any equality or identity in general, what about the case when $E(X_1)=0$ and $E(X_1X_2)=0$?