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Compact real form of a complex semisimple Lie algebra

Definition

Let $g$ be a finite-dimensional complex semisimple Lie algebra with Killing form $B$ (Killing form). A compact real form of $g$ is a real form $g_{0}$ of $g$ (Real form of a complex Lie algebra) whose Killing form $B_{g_{0}} = B ∣_{g_{0} \times g_{0}}$ is negative definite, that is $B (X, X) < 0$ for every nonzero $X \in g_{0}$ and $B (X, X) = 0$ if and only if $X = 0$ . A real Lie algebra is called compact when its Killing form is negative definite; under this terminology the compact real forms are exactly the real forms that are compact as real Lie algebras. The existence and conjugacy of compact real forms are the content of Existence of a compact real form and Conjugacy of compact real forms.

Depends on

Used by

Same complexification with different killing form signatures Counterexample
Compact and split real forms of sl two c Example
All real forms of a complex semisimple lie algebra are isomorphic False statement
Classical real forms of the classical complex lie algebras Proposition
Classification of real semisimple lie algebras Theorem
Conjugacy of compact real forms Theorem
Every real Cartan subalgebra is conjugate to a theta-stable one Theorem
Existence of a Cartan involution Theorem
Existence of a compact real form Theorem

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter VI (standard reference, not scraped)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19-24 (standard reference, not scraped)