rotation around the $z$ axis is generated by $$\label{dir15}
L_{3} = \pmatrix{0 & -i & 0 & 0 & 0 \cr i & 0 & 0 & 0 & 0 \cr
  0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 } .$$ The generators $L_1$ and $L_2$ can be also be constructed. The $K_3$ and $Q_3$ generators will take the form $$K_{3} = \pmatrix{0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
  0 & 0 & 0 & i & 0 \cr 0 & 0 & i & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 } , \qquad
Q_{3} =\pmatrix{0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 \cr
               0 & 0 & 0 & 0 & i \cr 0 & 0 & 0 & 0 & 0 \cr
               0 & 0 & i & 0 & 0 } .$$ From these two matrices, the generators $K_1, K_2, Q_1, Q_2$ can be constructed. 