DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Cartan decomposition of a real semisimple Lie algebra

Definition

Let $g_{0}$ be a finite-dimensional real semisimple Lie algebra with a Cartan involution $θ$ (Cartan involution of a real semisimple Lie algebra). The Cartan decomposition attached to $θ$ is the eigenspace decomposition

$g_{0} = k_{0} \oplus p_{0}, k_{0} = {X : θ X = X}, p_{0} = {X : θ X = - X},$

of $g_{0}$ into the fixed and anti-fixed subspaces of the involution. Both summands are real subspaces and the sum is direct because an element of $k_{0} \cap p_{0}$ satisfies $X = - X$ ; every $X$ decomposes as $X = \frac{1}{2} (X + θ X) + \frac{1}{2} (X - θ X)$ . The bracket relations and the signs of the Killing form are established in Bracket relations and Killing signs in a Cartan decomposition.

Depends on

Cartan involution of a real semisimple Lie algebra

Used by

Maximal split abelian subspace and real rank Definition
Riemannian symmetric pair of noncompact type Definition
Theta-stable Cartan subalgebras and their compact and split parts Definition
A nonreduced bc root system from a real form Example
Cartan involution and k plus p for sl n r Example
Compact and split cartan subalgebras of sl two r Example
Hyperbolic space as so zero n one mod so n Example
Iwasawa decomposition of sl two r Example
Polar cartan decomposition of sl n r Example
Restricted roots of sl n r Example
Bracket relations and Killing signs in a Cartan decomposition Proposition
Real Cartan subalgebras need not be conjugate Proposition
Global Cartan decomposition for a connected finite center semisimple Lie group Theorem
Maximal abelian subspaces of p are conjugate by K Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter VI (standard reference, not scraped)