quantum mechanics - Deriving the Pauli-Schrödinger equation from the Dirac equation

Since the Schrödinger Pauli equation describes a non-relativistic spin ½ particle. This equation must be an approximation of the Dirac equation in an electromagnetic field. I was trying to derive this but I got stuck at a point. The free particle Dirac can be reduced to the equations

$$\begin{align} σ^{i}(p_{i}+eA_{i}) u_B & = (E-m+eA_0)u_A. \\ \sigma^{i}(p_{i}+eA_{i})u_A & = (E+m+A_{0})u_B \end{align}$$

I multiplied both sides of the first equation by $(E+m+eA_0)$ to get the Schrödinger Pauli equation. I was not able to eliminate $u_B$ completely from the equation. Can someone help me with the derivation?