Indefinite symmetric spaces with \(G_{2(2)}\)-structure (Q2840171)

In contrast to the Riemannian situation, where homogeneous spaces with holonomy contained in \(G_2 \subset SO(7)\) are flat [\textit{D. V. Alekseevskii} and \textit{B. N. Kimel'fel'd}, Funct. Anal. Appl. 9, 97--102 (1975); translation from Funkts. Anal. Prilozh. 9, No. 2, 5--11 (1975; Zbl 0316.53041)], there exist indecomposable indefinite symmetric spaces whose holonomy is contained in the non-compact real form \(G_{2(2)} \subset SO(4, 3)\). For an overview on the general problem see [\textit{I. Kath} and \textit{M. Olbrich}, in: Recent developments in pseudo-Riemannian geometry. Zürich: European Mathematical Society. 1--52 (2008; Zbl 1158.53044)].NEWLINENEWLINEThis paper determines all indecomposable pseudo-Riemannian symmetric spaces of signature \((4, 3)\) with holonomy in \(G_{2(2)}\). The classification relies on the fact that the transvection group is solvable, so the associated symmetric triple \((\mathfrak{g}, \theta, \langle, \rangle)\) can be viewed as the quadratic extension of a Lie algebra with involution, as explained in [\textit{I. Kath} and \textit{M. Olbrich}, Transform. Groups 14, No. 4, 847--885 (2009; Zbl 1190.53051)].