Killing form
A Lie algebra has no obvious notion of length or angle, but it can manufacture one from its own bracket. Each element x acts on the algebra by bracketing (the map ad(x)); composing two such actions and taking the trace produces a number that depends symmetrically on the two elements. This canonical symmetric pairing, built purely from the algebra's internal structure, is the Killing form, and its degeneracy or nondegeneracy reads off deep structural facts.
Define the Killing form by B(x, y) = trace(ad(x) ad(y)), where ad(x)(z) = [x, z]. It is a symmetric bilinear form on L, and it is invariant (associative): B([x, y], z) = B(x, [y, z]). Invariance forces the radical of B to be an ideal of L, which is what links the form to the algebra's ideal structure.
Two Cartan criteria make the Killing form the central diagnostic tool (in characteristic zero). Solvability: L is solvable if and only if B(x, y) = 0 for all x in [L, L] and y in L. Semisimplicity: L is semisimple if and only if B is nondegenerate. So a single trace computation distinguishes the two extreme classes of Lie algebras.
For sl(2, C) with basis e, f, h, one computes B(h, h) = 8, B(e, f) = B(f, e) = 4, and all other pairings zero. The Gram matrix is nondegenerate (determinant nonzero), confirming sl(2, C) is semisimple.
A nondegenerate Killing form on sl(2, C) certifies semisimplicity.
The criteria require characteristic zero (and algebraic closure for the cleanest forms). In positive characteristic the Killing form can be degenerate even for simple Lie algebras, so it is not a reliable test there.