{i\over 4}\left(a^{\dag}a^{\dag} - aa + b^{\dag}b^{\dag} - bb\right) ,   \nonumber \\[3mm]
&{}& K_{3} = {1\over 2}\left(a^{\dag}b^{\dag} + ab\right) ,   \nonumber \\[3mm]
&{}& Q_{1} = -{i\over 4}\left(a^{\dag}a^{\dag} - aa - b^{\dag}b^{\dag} + bb \right) ,   \nonumber \\[3mm]
&{}& Q_{2} = -{1\over 4}\left(a^{\dag}a^{\dag} + aa + b^{\dag}b^{\dag} + bb \right) ,  \nonumber \\[3mm]
&{}& Q_{3} = {i\over 2}\left(a^{\dag}b^{\dag} - ab \right) .\end{aligned}$$ He then noted that these operators satisfy the following set of commutation relations. $$\begin{aligned}
\label{dir12}
&{}& [L_{i}, L_{j}] = i\epsilon _{ijk} L_{k} ,\qquad
[L_{i}, K_{j}] = i\epsilon _{ijk} K_{k} , \qquad
[L_{i}, Q_{j}] = i\epsilon _{ijk} Q_{k} , \nonumber\\[1ex]
&{}&
[K_{i}, K_{j}] = [Q_{i}, Q_{j}] = -i\epsilon _{ijk} L_{k} ,  \qquad
[L_{i}, S_{3}] = 0 ,  \nonumber\\[1ex]
&{}&
[K_{i}, Q_{j}] = -i\delta _{ij} S_{3} , \qquad
[K_{i}, S_{3}] =  -iQ_{i} , \qquad [Q_{i}, S_{3}] = iK_{i} .\end{aligned}$$

Dirac then determined that these commutation relations constitute the Lie algebra for the $O(3,2)$ de Sitter group with ten generators. This de Sitter group is the Lorentz group applicable to three space coordinates and two time coordinates. Let us use the notation $(x, y, z, t, s)$, with $(x, y, z)$ as space coordinates and $(t, s)$ as two time coordinates. Then the 