MinimalIdeal(S) calculates the smallest ideal $J$ containing the list of vectors S in an Lie algebra $\mathrm{\𝔤}$.

A list of vectors giving a basis for $J$is returned.

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

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

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[5\right]\right]\,\left[\left[\left[1\,5\,1\right]\,2\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,5\,2\right]\,1\right]\,\left[\left[2\,5\,3\right]\,1\right]\,\left[\left[3\,5\,3\right]\,1\right]\,\left[\left[4\,5\,4\right]\,2\right]\right]\right]\right)\:$

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

$\mathrm{MultiplicationTable}\left(''LieBracket''\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e5}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e2}\+\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e3}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=2\mathrm{e4}\right]$

$\mathrm{S1}≔\left[\mathrm{e1}\,\mathrm{e3}\right]\:$

$\mathrm{I1}≔\mathrm{MinimalIdeal}\left(\mathrm{S1}\right)$

$\mathrm{I1}≔\left[\mathrm{e1}\,\mathrm{e3}\right]$

$\mathrm{S2}≔\left[\mathrm{e2}\,\mathrm{e4}\right]\:$

$\mathrm{I2}≔\mathrm{MinimalIdeal}\left(\mathrm{S2}\right)$

$\mathrm{I2}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]$

$\mathrm{Query}\left(\mathrm{S2}\,''Ideal''\right)$

$\mathrm{false}$

$\mathrm{Query}\left(\mathrm{I2}\,''Ideal''\right)$

$\mathrm{true}$