It is a fact that if $\Lambda$ is a nonelementary subgroup of ${\rm PSL_2}(\mathbb{C})$ which contains an hyperbolic transformation and moreover ${\rm tr}(g)\in\mathbb{R}/\pm 1$ for all $g\in\Lambda$ then in fact $\Lambda$ is a Fuchsian subgroup, i.e. contained in a conjugate of ${\rm SL}_2(\mathbb{R})$. The proof I am aware of for this is rather unenligthening since it proceeds by simple algebraic manipulations on $2\times 2$ matrices.

Is there a more geometric proof of this? The main motivation for asking this is the following question, which is a tentative generalization of the above fact to higher dimensions: can a (torsion-free, say) nonelementary subgroup $\Lambda$ of ${\rm SO}(n,1)^\circ$ ($n$ odd) satisfy the condition that any of its semisimple elements has an holonomy (i.e. rotational part) which fixes a subspace of codimension $\ge 2$ in $\mathbb{R}^n$ and be Zariski-dense (if no, the proof would likely proceed by showing that it must preserve a totally geodesic subspace)?

For lattices there is enough holonomies so that the condition above is never satisfied: actually the holonomies becomes equidistributed in ${\rm SO}(n)$ as one goes through the hyperbolic conjugacy classes, by a result of P. Sarnak and M. Wakayama.

Edit: As remarked in Misha's answer the question was carelessly phrased for even-dimensional hyperbolic spaces; I added the oddness condition in the original formulation, for an even dimension $n$ the question becomes 'can a Zariski-dense subgroup of ${\rm SO}(n,1)$ contain only elements which fix a subspace of dimension $\ge 3$ in $\mathbb{R}^{n+1}$?'