How to rewrite the Lagrangian density for the Pauli moment in Dirac theory

I am trying to follow a derivation in a paper (DOI: 10.1119/1.2710486).

It defines the Lagrangian density for an ananomalous magnetic moment (Pauli moment) in the Dirac theory as,

$$mathcal{L}_{Pauli}=-kappa frac{mu_{B}}{2}bar{Psi}sigma^{munu}Psi F_{munu}$$

where $Psi$ is the Dirac field, $bar{Psi}$ the conjugate field, $sigma^{munu}=(i/2)(gamma^{mu}gamma^{nu}-gamma^{nu}gamma^{mu})$ and

$$F_{munu}=partial_{mu}A_{nu}-partial_{nu}A_{mu}=begin{pmatrix}
0 & pmb{mathscr{E}}_{x} & pmb{mathscr{E}}_{y} & pmb{mathscr{E}}_{z}
– pmb{mathscr{E}}_{x} & 0 & -textbf{B}_{z} & textbf{B}_{y}
– pmb{mathscr{E}}_{y} & textbf{B}_{z} & 0 & -textbf{B}_{x}
– pmb{mathscr{E}}_{z} & -textbf{B}_{y} & textbf{B}_{x} & 0
end{pmatrix}$$

is the electromagnetic tensor.

It then proceeds to rewrite $mathcal{L}_{Pauli}$ in terms of $pmb{mathscr{E}}$ and $textbf{B}$ fields to obtain:

$$mathcal{L}_{Pauli}=-kappa mu_{B}bar{Psi}[pmb{Sigma}cdottextbf{B}-ipmb{alpha}cdotpmb{mathscr{E}}]Psi$$

where,

$$pmb{Sigma}=begin{pmatrix}
sigma & 0
0 & sigma
end{pmatrix}, pmb{alpha}=begin{pmatrix}
0 & sigma
sigma & 0
end{pmatrix}$$

I can’t follow how they have rewritten the Lagrangian in this form. It is a long time since I have used this kind of maths so maybe I am missing something obvious. I thought that there could be a commutation rule I had forgotton about or something? Any help would be much appreciated.