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In this article, we determine the seven-dimensional almost Abelian Liealgebras which admit calibrated or parallel G_2-/G_2^*-structures. Along theway, we show that certain well-established curvature restrictions forcalibrated and parallel G_2-structures are not valid in the G_2^* case. In moredetail, we provide the first example of a Ricci-flat calibrated G_2^*-structureon a compact manifold whose holonomy is not contained in G_2^*. Moreover, weget examples of non-flat parallel G_2^*-structures on almost Abelian Liealgebras g. We give a full classification of these G_2^*-structures if g isadditionally nilpotent.

Calibrated and parallel structures on almost Abelian Lie algebras