Cartan’s criterion for solvability

A Lie algebra over characteristic 0 is solvable iff a certain trace pairing vanishes on g × [g,g].

Cartan’s criterion for solvability

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field of characteristic $0$ , and let $B$ be its Killing form :

B(X,Y)=\mathrm{tr}(\mathrm{ad}_X\circ \mathrm{ad}_Y).

Write $\mathfrak{g}'=[\mathfrak{g},\mathfrak{g}]$ for the derived subalgebra .

Theorem (Cartan’s criterion for solvability). $\mathfrak{g}$ is solvable if and only if

B(X,Y)=0 \quad \text{for all } X\in \mathfrak{g},\; Y\in \mathfrak{g}'.

Equivalently,

\mathrm{tr}(\mathrm{ad}_X\circ \mathrm{ad}_Y)=0 \quad \text{for all } X\in \mathfrak{g},\; Y\in [\mathfrak{g},\mathfrak{g}].

Motivation. Solvability is defined in terms of the derived series , but Cartan’s criterion replaces an iterative bracket computation with a single trace-vanishing condition. It is particularly effective when $\mathfrak{g}$ is presented as a subalgebra of $\mathfrak{gl}(V)$ , where traces can be computed concretely.

Remark. The criterion is compatible with the heuristic that “brackets measure noncommutativity”: the condition tests how far commutators act nontrivially through $\mathrm{ad}$ . Compare also Cartan’s semisimplicity criterion .