Subalgebras of Lie algebras with non-degenerate restriction of the Killing form

TL;DR

This paper proves that any subalgebra of a finite-dimensional Lie algebra with a non-degenerate Killing form is necessarily reductive, highlighting a key structural property of such subalgebras.

Contribution

The paper establishes that subalgebras with non-degenerate Killing forms are always reductive, providing a new insight into the structure of Lie algebras.

Findings

Subalgebras with non-degenerate Killing form are reductive.

The result applies to all finite-dimensional Lie algebras.

Provides a structural criterion for subalgebras based on the Killing form.

Abstract

Let $\mf{g}$ be any finite-dimensional Lie algebra with Killling form $B$. Let $\mf{h}$ be a subalgebra of $\mf{g}$ on which the Killing form is non degenerate. Then $\mf{h}$ is reductive.