In two dimensions a very nice parametrization of the rotation group is obtained by the following line of arguments:
- The group of rotations is connected and compact. Therefore the exponential is surjective.
- For every not null vector $v$ there is a unique vector $\star v$ such that $(v,\star v)$ is a positively oriented orthogonal basis and $\|\star v\|=\|v\|$. Then $\star v$ has the simple interpretation of being the counter-clockwise rotation by $90^\circ$ of $v$. One immediately identifies $\star$ as the Hodge-star operator and proofs it is an element of the Lie algebra.
- By a simple calculation $e^{\varphi\star}v=\cos(\varphi)v+\sin(\varphi)\star v$ for every $\varphi\in\mathbb{R}$.
A similar path can be taken for the three-dimensional rotation group through the element of its Lie algebra $\varphi\cdot J$ defined by $\varphi\cdot Jv=\varphi\times v:=\star(\varphi\wedge v)$ for all vectors v. The idea is similarly that $(\varphi, \varphi\times v, \varphi\times(\varphi\times v))$ is an oriented basis. This leads to Rodrigues' formula.
I would like to know if anybody knows of such a simple characterization for the (proper orthochronous) Lorentz group? Since every element of the Lorentz group can be written using the product of a rotation and a boost, I am particularly interested in the boosts. I would like in particular a geometrical and basis-free description. I seem to be able to retain such descriptions much better. I imagine that the problem statement would be something like
Consider a unit timelike vector $t$ (indicating the 4-velocity of the observer) and a unit spatial vector $n$ (indicating the preferred direction in space). Consider a Lorentz transformation $L$ which leaves invariant $\text{span}\{t,n\}^\bot$. One should then obtain a relation like Rodrigues' formula (see wikipedia) using a geometrically inspired Lie algebra element (I don't understand the geometric content of the usual $K$'s). The parameters multiplying this Lie algebra element should then be clearly identifiable with rapidity or some related parameter.