\begin{aligned}0&= ({p} + m c) \gamma_0 \gamma_0 ({p} - m c) u \\ &=\left(\frac{1}{{c}} \left( \mathcal{E} - e \phi \right)+\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)+ \gamma_0 m c\right)\left(\frac{1}{{c}} \left( \mathcal{E}- e \phi \right)-\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)- \gamma_0 m c \right) u \\ &=\frac{1}{{c}} \left( \mathcal{E} - e \phi \right)\left(\frac{1}{{c}} \left( \mathcal{E}- e \phi \right)-\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)- \gamma_0 m c \right) u \\ &\qquad +\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)\left(\frac{1}{{c}} \left( \mathcal{E}- e \phi \right)-\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)- \gamma_0 m c \right) u \\ &\qquad + \gamma_0 m c\left(\frac{1}{{c}} \left( \mathcal{E}- e \phi \right)-\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)- \gamma_0 m c \right) u \\ &=\left(\frac{1}{{c^2}} \left( \mathcal{E} - e \phi \right)^2- \left( \boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right) \right)^2- (mc)^2\right) u\\ &\qquad + \frac{1}{{c}} \left[{\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)},{\mathcal{E} - e \phi}\right] u- m c\left\{{\boldsymbol{\alpha} \cdot \left(\mathbf{P} - \frac{e}{c} \mathbf{A} \right)},{ \gamma_0}\right\} u \\ &\qquad + {\gamma_0 m\left(\mathcal{E} - e \phi\right) u}- {\gamma_0 m\left(\mathcal{E} - e \phi\right) u}\\ \end{aligned} 