Let \(\mathbb{R}^4_q\) be the \(4\)-dimensional pseudo-Euclidean space with index \(q\), that is, the real vector space \(\mathbb{R}^4\) equipped with the metric \[ \langle x ,y \rangle =-\sum_{i=1}^{q}x_iy_i+\sum_{i=q+1}^{4}x_iy_i, \] where \(x=(x_1,x_2,x_3,x_4)\), \(y=(y_1,y_2,y_3,y_4)\) and \(1\leqslant q\leqslant 3\). The Lorentzian submanifold \(\mathbb{S}^3_1=\{x\in\mathbb{R}^4_1|\langle x,x\rangle =1\}\) (resp., \(\mathbb{H}^3_1=\{x\in \mathbb{R}^4_2|\langle x,x\rangle=-1\}\)) is called the three-dimensional de Sitter space (resp., anti-de Sitter space). The authors use the notation \(M^3_c\) for these two spaces, explicitly \(M^3_1=\mathbb{S}^3_1\) and \(M^3_{-1}=\mathbb{H}^3_1\). To study the orientable surfaces immersed in \(M^3_c\), they consider a special type of such surfaces, \(L_1\)-\(2\)-type surfaces defined in the preliminaries section. The main theorem of the paper states that if \(M^2_s\) (\(s=0\) for the Riemannian case, and \(s=1\) for the Lorentzian case) is an orientable \(L_1\)-\(2\)-type surface immersed in \(M^3_c\), then \(M^2_s\) has constant principal curvatures if and only if \(M^2_s\) has constant second mean curvature. In order to get the classification results they give in Section 3 four examples of surfaces immersed in \(M^3_c\): totally umbilical surfaces, standard pseudo-Riemannian products, complex circle and B-scroll. Finally, they show that an \(L_1\)-\(2\)-type surface is either an open portion of a standard pseudo-Riemannian product, or a B-scroll over a null curve, or otherwise its mean curvature, its Gaussian curvature and its principal curvatures are all nonconstant.