The derived algebra of a Lie algebra $\mathrm{\𝔤}\mathit{ }$is the span of the set of vectors $\left[x\, y\right]$for all $x\,y\in \mathrm{\𝔤}$. The derived algebra is an ideal in $\mathrm{\𝔤}$.

DerivedAlgebra(LieAlgName) calculates the derived algebra of the Lie algebra $\mathrm{\𝔤}\mathit{ }$defined by LieAlgName. If no argument is given, then the derived algebra of the current Lie algebra is found.

DerivedAlgebra(S) calculates the derived algebra of the Lie subalgebra $S$ (viewed as a Lie algebra in its own right).

A list of vectors defining a basis for the derived algebra of $\mathrm{\𝔤}$ (or $S$) is returned. If the derived algebra is trivial, then an empty list is returned.

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

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

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[4\right]\right]\,\left[\left[\left[2\,4\,1\right]\,1\right]\,\left[\left[3\,4\,3\right]\,1\right]\right]\right]\right)$

$\mathrm{L1}≔\left[\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\right]$

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

$\mathrm{DerivedAlgebra}\left(\right)$

$\left[\mathrm{e1}\,\mathrm{e3}\right]$

$\mathrm{DerivedAlgebra}\left(\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e4}\right]\right)$

$\left[\mathrm{e1}\right]$