the Killing form
/ KILL-ing (after Wilhelm Killing) /
A Lie algebra has no built-in notion of length or angle; the bracket alone gives no inner product. The Killing form manufactures one canonically out of the bracket itself, by asking how the algebra acts on itself. It is the natural symmetric bilinear form attached to any Lie algebra, and reading off its signature and degeneracy diagnoses the deepest structural facts — whether the algebra is semisimple, compact, or has a center.
Precisely, on a finite-dimensional Lie algebra g define B(X, Y) = trace(ad(X) ad(Y)), where ad(X) = [X, -] is the adjoint endomorphism of g and the trace is of the composite linear map ad(X) ad(Y): g -> g. This B is symmetric and bilinear, and it is invariant (associative): B([X, Y], Z) = B(X, [Y, Z]), and Ad-invariant, B(Ad(g)X, Ad(g)Y) = B(X, Y). Two theorems make it a diagnostic instrument. Cartan's criterion for semisimplicity: g is semisimple if and only if B is nondegenerate (its matrix has nonzero determinant). Cartan's criterion for compactness/solvability flavor: a real semisimple g is the Lie algebra of a compact group if and only if B is negative definite. So you compute one trace form and learn whether the algebra is rigid (semisimple) and whether it integrates to a compact group.
The Killing form is the workhorse behind root systems, the classification of simple Lie algebras, and the construction of bi-invariant metrics on compact groups (negate B to get a positive-definite invariant inner product, hence a bi-invariant Riemannian metric). Honesty: the Killing form can be degenerate (it vanishes identically on any abelian Lie algebra, and is degenerate on any non-semisimple one), so it is NOT always a metric. And it is only ONE invariant form; semisimple algebras can carry other invariant bilinear forms, though on a SIMPLE algebra every invariant symmetric form is a scalar multiple of B.
For su(2), parametrize by Pauli-type generators with [e_1, e_2] = e_3 (and cyclically). Then ad(e_1) is the matrix sending e_2 |-> e_3, e_3 |-> -e_2, e_1 |-> 0, and one computes B(e_i, e_i) = trace(ad(e_i)^2) = -2 for each i, with B(e_i, e_j) = 0 for i != j. So B is negative definite — diagnosing su(2) as semisimple and compact-type, matching SU(2) being compact.
The Killing form of su(2) is negative definite, certifying compact, semisimple type.
The Killing form is degenerate (in fact zero) on any abelian Lie algebra, so it is not always an inner product. Nondegeneracy of B IS the definition-grade test for semisimplicity (Cartan's criterion).