Sliced Extensions, Irreducible Extensions, and Associated Graphs: An Analysis of Lie Algebra Extensions. II. Application to Euclidean, Poincaré, and Galilean Algebras

The results of a preceding paper on Lie algebra extensions and sliced extensions are applied to the Lie algebras $\mathcal{E}\text{(3),}\mathcal{P},\mathrm{and}\hspace{0.17em}\mathcal{G}$ of the Euclidean, resp. Poincaré and Galilean groups. The primitive extensions are analyzed in detail. A procedure for the construction of irreducible extensions is illustrated by some examples, using diagrams which picture the graphs of the extensions. It is proved that all extensions by $\mathcal{E}\text{(3),}\mathcal{P},\mathrm{and}\hspace{0.17em}\mathcal{G}$ are inessential.