On pseudo-Riemannian Lie algebras
On pseudo-Riemannian Lie algebras
In this paper, we show that ${\mathfrak g} {\mathfrak g}\subset {\mathfrak g}$ if ${\mathfrak g}$ is a pseudo-Riemannian Lie algebra with $C(\mathfrak g)\not =0$. Then we show that ${\mathfrak g}{\mathfrak g}\subset {\mathfrak g}$ when $\dim \mathfrak g=4$, which leads to the classification of the pseudo-Riemannian Lie algebras in dimension 4.