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Let A" be a real inner product space of finite or infinite dimension ^2, and let ρ φ Ο be a fixed real number. The following results will be presented in this note.A. A surjective mapping σ : X -» X preserving Lorentz-Minkowski distances 0 and Q in one direction must be a Lorentz transformation.B. The causal automorphisms of X, dim X ^ 3, are exactly the products δ λ, where λ is an orthochronous Lorentz transformation and δ a dilatation jc -> αχ, R 9 a > 0.C. If Q > 0, there exist A" and an injective σ : X -> X preserving Lorentz-Minkowski distance ρ, such that σ is not a Lorentz transformation. This result can be extended, mutatis mutandis, to Euclidean and Hyperbolic Geometry. It could be possible that Theorem 1 also holds true in the infinite-dimensional case provided that ρ < 0. However, a proof, if it exists, is not yet known. Result C shows that Theorem 1 cannot be extended to the infinite-dimensional case if ρ > 0, not even in the injective case.