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.