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The symmetric invariant bilinear form defined by the trace of the product of adjoint operators.
Let be a finite-dimensional Lie algebra over a field . Its Killing form is the symmetric bilinear form
Here is the adjoint representation. The form is ad-invariant.
For , with basis
one computes, using , that
For , the Killing form is
Over a field of characteristic , nondegeneracy of characterizes semisimplicity.
Let be a field. A Lie algebra over is a vector space over equipped with a -bilinear map
called the Lie bracket, such that and
The first condition says that the bracket is alternating; it implies . Over a field of characteristic different from , the two formulations are equivalent.
Let be a Lie algebra with Lie bracket . For , define a linear map
This gives a map
called the adjoint representation of .
Let be a finite-dimensional Lie algebra over a field of characteristic , and let be its Killing form:
Lemma (ad-invariance). For all ,
Equivalently, , i.e. each is skew-adjoint with respect to .
A finite-dimensional Lie algebra (over a field of characteristic ) is semisimple if it has no nonzero solvable ideals. Equivalently, if denotes the solvable radical, then is semisimple exactly when (compare solvable Lie algebra and ideals).
A fundamental characterization is:
Semisimple Lie algebras are the “non-abelian, non-solvable core” of general Lie algebras. The Levi decomposition expresses any finite-dimensional Lie algebra as a semidirect product of a semisimple Lie algebra with its solvable radical. Internally, every semisimple Lie algebra splits as a direct sum of simple ideals (see semisimple as a direct sum of simple ideals).
For complex semisimple Lie algebras, additional structure is encoded by roots (see roots of a Lie algebra) and the associated root system.