Supersymmetric geometries of IIA supergravity III

We find that (massive) IIA backgrounds that admit a G2⋉ℝ8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {G}_2\ltimes {\mathbb{R}}^8 $$\end{document} invariant Killing spinor must exhibit a null Killing vector field which leaves the Killing spinor invariant and that the rotation of the Killing vector field satisfies a certain g2 instanton condition. This result together with those in [4] and [5] complete the classification of geometries of all (massive) IIA backgrounds that preserve one supersymmetry. We also explore the geometry of a class of backgrounds which admit a G2⋉ℝ8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {G}_2\ltimes {\mathbb{R}}^8 $$\end{document} invariant Killing spinor and where in addition an appropriate 1-form bilinear vanishes. In all cases, we express the fluxes of the theory in terms of the geometry.