I am looking for a reference for the following folklore theorem, which is the Lorentzian analogue of the Bonnet fundamental theorem of surface in Euclidean space, hyperbolic space or $3$d sphere.



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> If a Coddazzi tensor $b$ on a Riemannian surface $(S,g)$ of curvature $K$ satisfies the Gauss-equation $K=-det b+ k$ then there exits a unique (up to global isometries) isometric embedding of $(S,g)$ in a Lorentzian space $M$ of constant curvature $k$, such that $b$ is the pull-back by the immersion of the shape operator.
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In the tome 4 of Spivak, the case when $M$ is Minkowski space is more or less given as an exercise, although it is not stated explicitly.



Reference for Bonnet Fundamental theorem of surfaces in Lorentzian spaces
See Section 7 (you have to scroll down a bit to get the corollary for the semi-Riemannian case) in 


*Christian Bär, Paul Gauduchon, and Andrei Moroianu*, [**Generalized cylinders in semi-Riemannian and Spin geometry**](http://dx.doi.org/10.1007/s00209-004-0718-0), *Math. Z.* **249** (2005), no. 3, 545--580.


where the theorem is proved for the hypersurface case in arbitrary dimensions. (So not just the 2D version as referred to in the OP.) 


A few caveats:


1. The theorem proves the existence of a *local* embedding around every point.
2. By patching you get the existence of a *global immersion* provided that the hypersurface is simply connected. (This part is standard and the authors just refer to Kobayashi-Nomizu.)
3. Coming back to the question as asked above, as a side remark note that a closed Riemann surface cannot be isometrically embedded in $\mathbb{R}^{1,2}$, for the reason that any closed hypersurface of $\mathbb{R}^{1,2}$ must have a point where the tangent plane is time-like. The same is true for embeddings in to anti-de-Sitter. This is of course different from the Euclidean setting.


