  We give a generalized Weierstrass formula for a Lorentz surface conformally
immersed in the four-dimensional space $\mathbb{R}^{2,2}$ using spinors and
Lorentz numbers. We also study the immersions of a Lorentzian surface in {\bf
the} Anti-de Sitter space (a pseudo-sphere in $\mathbb{R}^{2,2}$): we give a
new spinor representation formula and deduce the conformal description of a
flat Lorentzian surface in that space.
