How to show that b1 and SSE are independent?

#1
under normal sampling zero covariance implies independence.
So I tried to solve COV(b1,SSE).
COV(b1,SSE) = E[(b1-beta1)(SSE-sigma^2*(n-2))]
=...
=E(b1*SSE)-beta1*sigma^2*(n-2)
But I don't know how to solve E(b1*SSE).
Do you know how to solve this?
Or did I do something wrong?:confused::(