On the Killing form on Lie algebras

Finite-dimensional Lie algebras, whose Killing form is trivial, are studied. The main properties of the form Killing are specified, a series of examples of Lie algebras with trivial or non-trivial Killing form are given. It is given a classification of Lie algebras of dimension $\le 5$ with trivial Killing form. Some results are found about the space of all Lie algebras with the trivial Killing form.