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 $${\mathrm {G}}_2^{*}$$.

中文翻译：

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在 $${\mathrm {G}}_2^*$$G2∗ 中具有完整度的局部 II 型度量

Fino 和 Kath 提供了包含在例外的非紧李群 $${\mathrm {G}}_2^{*}$$ 中的具有不可分解的完整表示的可能完整群的列表。分类是由于相应的完整代数而分为I型、II型和III型，具体取决于鞋底的维数分别为1、2或3。Fino 和 Kath 还表明，所有类型 I 的代数，以及所有类型 III 的代数确实可以通过带有签名 (4, 3) 的度量实现为完整代数。这篇文章证明这对于所有类型 II 代数也是如此。因此，对于包含在 $${\mathrm {G}}_2^{*}$$ 中的所有不可分解的完整群，存在一个度量的实现。