Denote by $\text{SO}\_0(1,4)$ the identity component of the special linear isometry group $\mathrm{SO}(1,4)$ of the Lorentz-Minkowski space $\mathbb{R}\_1^5$, that is, of
$$\text{SO}(1,4)=\left\{X\in\text{SL}(5,\mathbb{R})\mid X^tI\_{1,4}X=I\_{1,4}\right\},\quad\text{where}\;I\_{1,4}:=\text{diag}(-1,1,1,1,1).$$
Is there any explicit matrix expression for a general element of this Lie group?
Maybe using quaternions or spin representation?



Matrix expression for elements of $\text{SO}_0(1,4)$
The [Cartan decomposition](https://en.wikipedia.org/wiki/Cartan_decomposition#Cartan_decomposition_on_the_Lie_group_level):
$$
\mathrm{SO}\_0(1,4)=\left\{\left(
\begin{array}{c|c}
1&0\\\hline
0&A
\end{array}\right)
\exp\left(\begin{array}{c|c}
0&{}^tb\\\hline
b&0
\end{array}\right): A\in\mathrm{SO}(4), b\in\mathbf R^{4\times1}
\right\},
$$
(where the exponential is easily computed explicitly in terms of hyperbolic sines and cosines) boils this down to your favorite parametrization of [$\mathrm{SO}(4)$](https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space).


