K dot p method with SO interaction

[aaaa202]

I am trying to derive the k dot p formula with the spin orbit interaction term included. I am having some trouble. I insert a bloch wave function multiplied with an unspecified spinor. But how do I treat the cross product term that comes from the SO and what is the effect of the pauli matrix vector on the spinor? And also: Since the spin orbit term contain the gradient of the coulomb potential how can I let p act on the bloch states? (I assume this how you derive the correct formula?)
The result I want is the attached. Also I attached what I have written up so far.

 

[DrDu]

This is a problem of degenerate perturbation theory. I would first try to diagonalize the sigma matrices.