Local Type II metrics with holonomy in $${\mathrm {G}}_2^*$$

Abstract

A list of possible holonomy groups with indecomposable holonomy representation contained in the exceptional, non-compact Lie group $${\mathrm {G}}_2^{*}$$ was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1, 2 or 3, respectively. It was also shown by Fino and Kath that all algebras of Type I, and by the author that all of Type III are indeed realizable as holonomy algebras by metrics with signature (4, 3). This article proves that this is also true for all Type II algebras. Thus, there exists a realization by a metric for all indecomposable holonomy groups contained in $${\mathrm {G}}_2^{*}$$.