From the abstract: We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold \(M_{c,r}^m\), of dimension \(m\), constant sectional curvature \(c \neq 0\), and signature \(r\), into a pseudo-Euclidean space \(\mathbb{R}_s^{m+k}\), of signature \(s \geq r\). In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space \(S_s^{m+k}\) or \(H_s^{m+k}\), which have been known for some time. A simple formula is given for obtaining these immersions from those loop group maps.