In three-dimensional spacetime, there are two kinds of slow energy-momentum vectors:
$$ \begin{bmatrix} \pm m \cosh{(\xi)} & m \sinh{(\xi)} \cos{(\theta)} & m \sinh{(\xi)} \sin{(\theta)} \end{bmatrix}. $$
Similarly, there are two kinds of null energy-momentum vectors:
$$ \begin{bmatrix} \pm k & k \cos{(\theta)} & k \sin{(\theta)} \end{bmatrix}. $$
However, there is only one kind of fast energy-momentum vectors:
$$ \begin{bmatrix} w \sinh{(\rho)} & w \cosh{(\rho)} \cos{(\theta)} & w \cosh{(\rho)} \sin{(\theta)} \end{bmatrix}. $$
In all three examples, you have
$$ m > 0, \qquad k > 0, \qquad w > 0, \qquad -\infty < \xi < \infty, \qquad -\infty < \rho < \infty, \qquad -\pi < \theta \leq \pi. $$
You can construct Lorentz invariants by looking at the inner product of two energy-momentum vectors.
Slow and Slow
The inner product of two slow energy-momentum vectors is:
$$ P_{1} \cdot P_{2} = -m_{1} m_{2} \left[ \cosh{(\xi_{1})} \cosh{(\xi_{2})} - \sinh{(\xi_{1})} \sinh{(\xi_{2})} \cos{(\theta_{1} - \theta_{2})} \right]. $$
Note that
$$ \cos{(\theta_{1} - \theta_{2})} = \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} - \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)}; $$
and
$$ 1 = \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} + \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)}. $$
Thus,
$$ P_{1} \cdot P_{2} = - m_{1} m_{2} \left[ \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \cosh{(\xi_{1} - \xi_{2})} + \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \cosh{(\xi_{1} + \xi_{2})} \right]. $$
Slow and Null
The inner product of one slow and one null energy-momentum vector is:
$$ P_{1} \cdot P_{2} = -m_{1} k_{2} \left[ \cosh{(\xi_{1})} - \sinh{(\xi_{1})} \cos{(\theta_{1} - \theta_{2})} \right]. $$
This can also be written as
$$ P_{1} \cdot P_{2} = -m_{1} k_{2} \left[ \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \exp{(-\xi_{1})} + \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \exp{(\xi_{1})} \right]. $$
Slow and Fast
The inner product of one slow and one fast energy-momentum vector is:
$$ P_{1} \cdot P_{2} = -m_{1} w_{2} \left[ \cosh{(\xi_{1})} \sinh{(\rho_{2})} - \sinh{(\xi_{1})} \cosh{(\rho_{2})} \cos{(\theta_{1} - \theta_{2})} \right]. $$
This can also be written as
$$ P_{1} \cdot P_{2} = m_{1} w_{2} \left[ \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \sinh{(\xi_{1} - \rho_{2})} - \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \sinh{(\xi_{1} + \rho_{2})} \right]. $$
Null and Null
The inner product of two null energy-momentum vectors is
$$ P_{1} \cdot P_{2} = - k_{1} k_{2} \left[ 1 - \cos{(\theta_{1} - \theta_{2})} \right]. $$
This can also be written as
$$ P_{1} \cdot P_{2} = - 2 k_{1} k_{2} \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)}. $$
Null and Fast
The inner product of one null and one fast energy-momentum vector is
$$ P_{1} \cdot P_{2} = -k_{1} w_{2} \left[ \sinh{(\rho_{2})} - \cosh{(\rho_{2})} \cos{(\theta_{1} - \theta_{2})} \right]. $$
This can also be written as
$$ P_{1} \cdot P_{2} = k_{1} w_{2} \left[ \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \exp{(-\rho_{2})} - \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \exp{(\rho_{2})} \right]. $$
Fast and Fast
The inner product of two fast energy-momentum vectors is
$$ P_{1} \cdot P_{2} = -w_{1} w_{2} \left[ \sinh{(\rho_{1})} \sinh{(\rho_{2})} - \cosh{(\rho_{1})} \cosh{(\rho_{2})} \cos{(\theta_{1} - \theta_{2})} \right]. $$
This can also be written as
$$ P_{1} \cdot P_{2} = w_{1} w_{2} \left[ \cos^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \cosh{(\rho_{1} - \rho_{2})} - \sin^{2}{\left( \frac{\theta_{1} - \theta_{2}}{2} \right)} \cosh{(\rho_{1} + \rho_{2})} \right]. $$
