# Generic parametrization of Lorentz transformation matrix?

A proper Lorentz transformation of a vector $$\bf{x}$$ is given by

$$\bf{x}\to \bf{x}'=\Lambda\cdot\bf{x}$$

where $$\Lambda$$ is a matrix with the properties

$$\Lambda^T\cdot\eta\cdot\Lambda=\eta~~~,~~~\det\Lambda=1,$$

where $$\eta=\text{diag}(-1,1,1,1)$$ is the Minkowski metric.

What is a convenient parametrization of all the individual components $$\Lambda_{ij}$$ of matrix $$\Lambda$$ for a generic proper Lorentz transformation continuously connected to the identity?

The usual six parameters for proper Lorentz transformations are a three-component axis-angle vector $$\vec\theta$$ for the rotation part and a three-component rapidity vector $$\vec\zeta$$ for the boost part, as described in this Wikipedia article.
If $$\vec J$$ are the rotation generators and $$\vec K$$ the boost generators, then
$$\Lambda(\vec\theta,\vec\zeta)=e^{\vec\theta\cdot\vec J-\vec\zeta\cdot\vec K}.$$