The Killing form on a Lie algebra g is the symmetric quadratic form defined by Killing(x, y) = trace(ad(x).ad(y)) for any x, y in g.  Here ad(x) and ad(y) are the ad matrices for the vector x and y.

In terms of the structure constants C^k_{ij} with respect to the basis e_i for g, Killing(e_i, e_j) = C^k_{il} C^l_{jk} (summation on l).

Killing() calculates the Matrix [Killing(e_i, e_j)] = [C^k_{il} C^l_{jk}], where the C^k_{ij} are the structure constants for the current Lie algebra.

Killing(Alg) calculates the Killing Matrix for the Lie algebra Alg.

Killing(h) calculates the Killing Matrix restricted to the subalgebra h.

The command Killing is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Killing(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Killing(...).