nLab
compact Lie algebra

Definition

A semisimple Lie algebra $\mathfrak{g}$ is compact if its Killing form $\langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is a negative definite bilinear form.

Properties

Proposition

The Lie algebra of a compact semisimple Lie group is compact.

A proof is spelled out for instance in (Woit, theorem 1).

References

For instance

• Peter Woit, Topics in Representation Theory: The Killing Form, Reflections and Classification of Root Systems (pdf)