CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Every finite-dimensional characteristic-zero Lie algebra is a matrix Lie algebra

Statement

Every finite-dimensional Lie algebra over a characteristic-zero field is isomorphic to a Lie subalgebra of ${g l}_{n} (k)$ for some finite $n \geq 0$ .

Facts & Assumptions

Given: Such a Lie algebra $g$ .

[L1]

Ado supplies a faithful finite-dimensional representation (Ado's theorem with nilpotent nilradical action).

[L2]

Faithfulness is injectivity, and a representation kernel is an ideal (Representation kernels are ideals).

Proof

technique · identify the algebra with its image

1.1

Choose the representation $ρ : g \to g l (V)$ from [L1] and put $n = \dim V$ . By [L2], $ρ$ is injective.

L1L2

2.1

Since $ρ$ preserves brackets, it is an isomorphism from $g$ to the Lie subalgebra $ρ (g) \subseteq {g l}_{n} (k)$ . If $g = 0$ , Ado allows $V = 0$ and $n = 0$ ; one may instead take the zero subalgebra of ${g l}_{1} (k)$ if positive matrix size is preferred.

step 1.1algebra∎

Depends on

Used by

Lie's third fundamental theorem Theorem

Dependency tree · two levels

17 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

Knapp, Lie Groups Beyond an Introduction, Theorem B.8 (standard reference, not scraped)