In Section 1 we characterize hyperbolic hyperplanes of dimension–free hyperbolic geometry over the real inner product space X of arbitrary (finite or infinite) dimension greater than 1 by euclidean hyperplanes ∋ 0 of \(X {\otimes} \mathbb{R}\) intersecting the surface \(\{(x,\sqrt{1 + x^2}) \mid x \in X\}\). This is for \(X = {\mathbb{R}}^2\) a well-known result of the classical theory, since the Weierstras model of plane hyperbolic geometry defines hyperbolic lines via the euclidean planes of \({\mathbb{R}}^3\) through 0 intersecting the surface in question. – In Section 2 formulas will be derived representing dimension–free hyperbolic motions of X as well as products consisting of two such factors. – Finally, in Section 3, coordinates will be considered describing naturally the action of hyperbolic translations on points.