The complexification of a real Lie algebra g of dimension n is a real Lie algebra of dimension 2n.  If e1, e2, ..., en is a basis for g, then e1, e2, ..., en, Ie1, Ie2, ..., Ien, where I^2 = - 1, is a basis for the complexification of g.

Complexify(AlgName1, AlgName2) calculates the complexification of the Lie algebra g defined by AlgName1.

A Lie algebra data structure is returned for the complexified Lie algebra with name AlgName2.  A Lie algebra data structure contains the structure constants of a Lie algebra in a standard format used by the LieAlgebras package.

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