negative definite (or negative semidefinite if the Lie algebra is not semisimple). Note that this is one of two inequivalent definitions commonly used for compactness of a Lie algebra; the other states that a Lie algebra is compact if it corresponds to a compact Lie group. The definition of compactness in terms of negative definiteness of the Killing form is more restrictive, since using this definition it can be shown that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups. If is a semisimple Lie algebra over the complex numbers, then there are several non-isomorphic real Lie algebras whose complexification is , which are called its real forms. It turns out that every complex semisimple Lie algebra admits a unique (up to isomorphism) compact real form . The real forms of a given complex semisimple Lie algebra are frequently labeled by the positive index of inertia of their Killing form. For example, the complex special linear algebra has two real forms, the real special linear algebra, denoted , and the special unitary algebra, denoted . The first one is noncompact, the so-called split real form, and its Killing form has signature . The second one is the compact real form and its Killing form is negative definite, i.e. has signature . The corresponding Lie groups are the noncompact group of real matrices with the unit determinant and the special unitary group , which is compact. Trace forms Let be a finite-dimensional Lie algebra over the field , and be a Lie algebra representation. Let be the trace functional on . Then we can define the trace form for the representation as Then the Killing form is the special case that the representation is the adjoint representation, . It is easy to show that this is symmetric, bilinear and invariant for any representation . If furthermore is simple and is irreducible, then it can be shown where is the index of the representation. See also Casimir invariant Killing vector field Citations References Lie groups Lie algebras