Let X be some random variable with unconditional mean 0.
Suppose I have that
$E[XY]=E[X|Y]=0$ and $E[XZ] \neq 0$
Does it follow that $E[XYZ]=0$?
Here are my thoughts:
$E[XYZ]=E[E[XYZ|Y,Z]]=E[ZE[XY|Y]|Z]=0$
where $E[XY|Y]=0$ follows from our assumptions