Would it help if I explained the question a little more?
The CEF is the conditional expectation function, i.e., the expectation of the conditional distribution of Y given X, and the BLP is the best linear prediction of the CEF, i.e., the function that minimizes , where .
In my workings, I've taken advantage of the law of iterated expectations (i.e., the marginal expectation of Y is the expectation of its conditional expectation), and the iterated product law (i.e., the expected product of X and Y is equal to the expected product of and the conditional expectation of Y given X).
I suppose the thing that is confusing me, is that the parameters of the BLP (alpha and beta) are functions of random variables, so are random variables (right?), so should I really be pulling them outside the expectations operator and treating them as constants?