The complexification of a real Lie algebra $\mathrm{\𝔤}$ of dimension $n$ is a real Lie algebra of dimension $2n$.  If $\left\{e_1\, e_2\, ..\.\, e_n\right\}$is a basis for $\mathrm{\𝔤}$, then $\left\{e_1\, e_2\, ..\. \, e_n\, {\mathrm{Ie}}_1\, {\mathrm{Ie}}_2\, ..\. \, {\mathrm{Ie}}_n\right\}$, where $I^2 \= -1$, is a basis for the complexification of $\mathrm{\𝔤}$,

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. The structure equations for the complexification are displayed. (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(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[3\right]\right]\,\left[\left[\left[1\,3\,1\right]\,1\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,3\,2\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{L2}≔\mathrm{Complexify}\left(\mathrm{Alg1}\,\mathrm{Alg2}\right)$

$\mathrm{L2}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e6}\right]\=\mathrm{e4}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\+\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e6}\right]\=\mathrm{e4}\+\mathrm{e5}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-\mathrm{e4}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=-\mathrm{e4}-\mathrm{e5}\,\left[\mathrm{e4}\,\mathrm{e6}\right]\=-\mathrm{e1}\,\left[\mathrm{e5}\,\mathrm{e6}\right]\=-\mathrm{e1}-\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)\:$

$\mathrm{Query}\left(\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\,\left[\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\right]\,''SymmetricPair''\right)$

$\mathrm{true}$