Transitive Lie Algebras

The connected Lie subgroups of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ Gl(n,\mathbb{R})\ $$\end{document} that are transitive on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \mathbb{R}^n_{\ast}\ $$\end{document} are, loosely speaking, canonical forms for the symmetric bilinear control systems of Chapter 2 and are important in Chapter 7. Most of their classification was achieved by Boothby [32] at the beginning of our long collaboration. The corresponding Lie algebras g (also called transitive because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \frak{g}x=\mathbb{R}^n\ $$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ x\in \mathbb{R}^n_{\ast}\ $$\end{document} are discussed and a corrected list is given in Boothby-Wilson [32]; that work presents a rational algorithm, using the theory of semisimple Lie algebras, that determines whether a generated Lie algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \{A,B\}_{\mathcal{L}}\ $$\end{document} is transitive.