Consider the Poisson brackets (symplectic structure) given by the Lorentz algebra (Lie algebra of $SO(1,3)$) $$\{M^{AB},M^{CD}\} \equiv \omega_{AB,CD}\mathrm{d}M^{AB} \mathrm{d} M^{CD} = \eta^{AC} M^{BD} - \eta^{AD} M^{BC} + \eta^{BD} M^{AC} - \eta^{BC} M^{AD}\,, $$ where $A,B,C,D$ range from 0 to 3, $M^{AB} =-M^{BA}$ are some coordinates on phase space, and $\omega_{AB,CD}$ are components of the symplectic form. The Minkowski tensor $\eta^{AB}$ has the non-zero components $\eta^{00} = -1$, $\eta^{11} = \eta^{22} = \eta^{33} = 1$.
Now this algebra has two Casimir elements $M^{A B} M^{CD} \eta_{AC} \eta_{BD}$ and $M^{AB} M^{CD} \epsilon_{ABCD}$, where $\epsilon_{ABCD}$ is the Levi-Civita symbol. This means that symplectic form is degenerate with respect to these two coordinates (i.e., it is strictly speaking a presymplectic form) and from the total of 6 independent components of $M^{AB}$ only 4 will be non-degenerate directions of the symplectic form.
In the rest of the sub-space there should be a set of four canonical coordinates $p_\phi,\phi,p_\theta,\theta$ so that at least locally $$\mathbf{\omega} = \mathrm{d} p_\phi \wedge \mathrm{d} \phi + \mathrm{d} p_\theta \wedge \mathrm{d} \theta \,.$$
How can I find such a set of coordinates?
One trick which is used in classifying the representations of the Lorentz group is to complexify the algebra and define the two vectors $$V_{k(\pm)} = \frac{1}{2} ( M^{ij}\epsilon_{ijk} \pm i M^{0 k})$$ with $i,j,k = 1,2,3$ and Einstein summation is assumed. The two vectors then commute $$\{V_{i(+)}, V_{j(-)}\} = 0$$ and they have the canonical $SO(3)$ commutation relations between their components $$\{V_{i(\pm)}, V_{j(\pm)}\} = \epsilon_{ijk} V_{k (\pm)}$$ $\sum_i (V_{i(\pm)})^2$ are then Casimir elements which correspond to combinations of the two Casimir elements mentioned above. Local canonical coordinates are then for instance $V_{3(\pm)}, \varphi_{(\pm)} \equiv \arctan(V_{1(\pm)}/V_{2(\pm)})$ and the symplectic form can be expressed locally as $$\omega = \mathrm{d} V_{3 (+)} \wedge \mathrm{d} \varphi_{(+)} + \mathrm{d} V_{3 (-)} \wedge \mathrm{d} \varphi_{(-)}$$ However, my problem with this is that these variables are complex, and we have thus doubled the size of our phase space to make this transformation. Hence, I am looking for a description without complexification. The Lorentz algebra is such an important algebra for theoretical physics, yet I have not found any reference and so far did not have much luck in finding a set of real canonical coordinates myself.