10.2 Lie Algebra

The Lie algebra $\mathrm{\mathfrak{sim}}(3)$ has seven generators:

$$G_{1}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}, G_{2}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}, G_{3}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}$$ (132)

$$G_{4}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}, G_{5}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)}, G_{6}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 \end{array}\right)},$$ (133)

$$G_{7}={\scriptstyle \left(\begin{array}{ccc\vert c} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & -1 \end{array}\right)}$$ (134)