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

## TL;DR

This paper proves that all Type II holonomy algebras contained in the exceptional Lie group G_2* can be realized by metrics with signature (4,3), completing the classification of such holonomy groups.

## Contribution

It demonstrates that all Type II algebras associated with G_2* holonomy are realizable by explicit metrics, extending previous results for Types I and III.

## Key findings

- All Type II holonomy algebras are realizable by metrics.
- Completes the classification of G_2* holonomy groups.
- Provides explicit metrics for Type II cases.

## Abstract

A list of possible holonomy groups contained 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 be realizable as a 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 holonomy groups contained in $\mathrm{G}_2^{*}$.