Conditional expectation $E(XY\mid Z)$

Question

I'm trying to solve the following problem: let $X$ and $Y$ be 2 independent standard normal random variables and let be $Z=X+Y$. Calculate $E(XY\mid Z)$. I tried many approaches, but without getting the result. Any suggestion? We know previously that $Z=X+Y$ and $W=X-Y$ are independent and that $E(X\mid Z)=\frac{1}{2}Z$.

Answer 1

Hint: $XY=\frac{1}{4}((X+Y)^2-(X-Y)^2)$, and expectation is linear. Recall that if $X$ and $Y$ are independent normal with the same distribution, then $X+Y$ and $X-Y$ are independent.