On random choice of elliptic and hyperbolic rotations of the Lorentz spaces

Elliptic and hyperbolic rotations of the $(n+1)$-dimensional Lorentz space can be represented as exponential of rank $2$ matrices of the real Lie algebra $\mathfrak{so}(1, n)$. We shown that the ratio of the volumes of the corresponding sets of matrices Euclidean norm $\leqslant r$ is equal to $(\sqrt2)^{n-1}-1$ for all $r > 0$. Consequently the portion of hyperbolic rotations near identity decreases exponentially with increasing $n$. Another corollary is that in case of Minkovski space of special relativity choose of elliptic and hyperbolic rotations near identity is equiprobable.