  CFTs in Euclidean signature satisfy well-accepted rules, such as the
convergent Euclidean OPE. It is nowadays common to assume that CFT correlators
exist and have various properties also in Lorentzian signature. Some of these
properties may represent extra assumptions, and it is an open question if they
hold for familiar statistical-physics CFTs such as the critical 3d Ising model.
Here we consider Wightman 4-point functions of scalar primaries in Lorentzian
signature. We derive a minimal set of their properties solely from the
Euclidean unitary CFT axioms, without using extra assumptions. We establish all
Wightman axioms (temperedness, spectral property, local commutativity,
clustering), Lorentzian conformal invariance, and distributional convergence of
the s-channel Lorentzian OPE. This is done constructively, by analytically
continuing the 4-point functions using the s-channel OPE expansion in the
radial cross-ratios $\rho, \bar{\rho}$. We prove a key fact that $|\rho|,
|\bar{\rho}| < 1$ inside the forward tube, and set bounds on how fast $|\rho|,
|\bar{\rho}|$ may tend to 1 when approaching the Minkowski space.
  We also provide a guide to the axiomatic QFT literature for the modern CFT
audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for
Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them.
We also review a classic result of Mack about the distributional OPE
convergence. Some of the classic arguments turn out useful in our setup. Others
fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable
Euclidean assumptions (OS theorem).
