Consider a frame $S'$ (of coordinates $x'$) connected to a frame $S$ (of coordinates $x$) through a proper orthocronous Lorentz transformation $\Lambda$ ($x' = \Lambda x$). Except for the case where $\Lambda$ is a pure rotation or the identity, frame S' is moving with respect to S, with a constant velocity $v$.
Is there a way to extract the velocity $\mathbf{v}$ from the matrix $\Lambda$ alone?
For the case where $\Lambda$ is a pure Lorentz boost ($\Lambda = B(\mathbf{v})$), then it can be written in terms of $\mathbf{v}$ as
$$ B(\mathbf{v}) = \begin{bmatrix} \gamma & -\frac{\gamma}{c}\mathbf{v}^T \\ -\frac{\gamma}{c}\mathbf{v} & 1\!\!1 + \frac{\gamma^2}{\gamma+1}\frac{\mathbf{v}\mathbf{v}^T}{c^2} \end{bmatrix} $$
so it's easy to extract the vector $\mathbf{v}$ from the matrix $\Lambda$. However, the most general element of the proper orthocronous Lorentz group is not a pure boost (but can be decomposed as a product of a boost and a rotation). For these cases, how can I obtain $\mathbf{v}$?
