The Killing form and Cartan's criteria

Define B(x,y) as the trace of ad(x)ad(y). Cyclicity of trace makes it symmetric, and the Jacobi identity gives ad([x,y])=[ad(x),ad(y)]. Combining these identities proves invariance: B([x,y],z)=B(x,[y,z]). Thus the form is independent of a chosen matrix realization of g. A defining-representation trace form can be proportional to the Killing form in important examples, but it is not automatically the same normalization or even nondegenerate for an arbitrary representation.

The radical of a bilinear form consists of vectors pairing to zero with every vector. Invariance makes the Killing radical an ideal. A matrix representing B is invertible exactly when this radical is zero. One vector with B(x,x)=0 is merely isotropic; it need not pair to zero with other vectors. Over ℂ, even a nondegenerate symmetric form has isotropic vectors in many dimensions, so positivity is the wrong diagnostic. Hermitian positivity belongs to a different type of form.

Cartan's semisimplicity criterion states that g is semisimple if and only if its Killing form is nondegenerate, for finite-dimensional Lie algebras over a characteristic-zero field; here the field is ℂ. Cartan's solvability criterion instead states that g is solvable if and only if B(g,[g,g])=0. The second condition does not mean the whole Killing form is zero. A solvable algebra can have a nonzero trace pairing on elements outside its derived algebra, as the two-dimensional affine example shows.

These are structural theorems, not consequences of matrix symmetry alone. A singular Killing matrix rules out semisimplicity but does not by itself prove that the whole algebra is solvable; an algebra with a semisimple factor and nonzero center supplies a counterexample. To prove solvability by the criterion, check orthogonality of the entire derived algebra to all of g. Likewise a nonzero determinant proves nondegeneracy in a chosen basis, but the value of that determinant changes under basis rescaling while the structural conclusion does not.

For sl(n,C), the Killing form is 2n times the ordinary trace of a product in the defining n-dimensional representation. This formula permits efficient calculations once its normalization is known. On a simple complex Lie algebra every invariant symmetric bilinear form is a scalar multiple of the Killing form. On a semisimple direct sum, separate simple factors may be scaled independently. Nondegeneracy survives multiplication by a nonzero scalar, but a dual-basis construction such as a Casimir element changes inversely with that scale.

Restriction to a compact real form gives a negative-definite Killing form, so its negative defines a useful positive real inner product there. The same assertion is false on arbitrary noncompact real forms, and it is not a positivity assertion on the complex vector space. Keeping the field and type of pairing explicit prevents a common transfer error between compact-group geometry and complex representation theory. This lesson uses the complex bilinear form to analyze algebraic structure, leaving positive Hermitian forms on representations to their separate context.