The radical of a Lie algebra $\mathrm{\𝔤}\mathit{ }$is the largest solvable ideal contained in $\mathrm{\𝔤}$. The radical of $\mathrm{\𝔤}\mathit{ }$can be calculated as the orthogonal complement of the derived algebra $\mathrm{\𝔤}\mathit{\'}$ of $\mathrm{\𝔤}\mathit{ }$with respect to the Killing form $B$, that is, rad$\left(\mathrm{\𝔤}\right)$ = $\{x \in \mathrm{\𝔤}\mathit{ }\mathit{\|}\mathit{ }B\left(x\, y\right) \= 0$for all $y \in \mathrm{\𝔤}\mathit{\'}\mathit{\}}$. See, for example, Fulton and Harris Representation Theory, Graduate Texts in Mathematics 129, Springer 1991, Proposition C.22 page 484.

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

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

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

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

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[7\right]\right]\,\left[\left[\left[1\,2\,2\right]\,2\right]\,\left[\left[1\,3\,3\right]\,-2\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[1\,4\,4\right]\,1\right]\,\left[\left[1\,5\,5\right]\,-1\right]\,\left[\left[2\,5\,4\right]\,1\right]\,\left[\left[3\,4\,5\right]\,1\right]\,\left[\left[4\,5\,6\right]\,1\right]\,\left[\left[4\,7\,4\right]\,1\right]\,\left[\left[5\,7\,5\right]\,1\right]\,\left[\left[6\,7\,6\right]\,2\right]\right]\right]\right)$

$\mathrm{L1}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=2\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e4}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=-\mathrm{e5}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e4}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e5}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e6}\,\left[\mathrm{e4}\,\mathrm{e7}\right]\=\mathrm{e4}\,\left[\mathrm{e5}\,\mathrm{e7}\right]\=\mathrm{e5}\,\left[\mathrm{e6}\,\mathrm{e7}\right]\=2\mathrm{e6}\right]$

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

$\mathrm{rad}≔\mathrm{Radical}\left(\right)$

$\mathrm{rad}≔\left[\mathrm{e7}\,\mathrm{e6}\,\mathrm{e5}\,\mathrm{e4}\right]$

$\mathrm{Query}\left(\mathrm{rad}\,''Solvable''\right)$

$\mathrm{true}$

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

$\mathrm{true}$

$A≔\left[\mathrm{e1}\,\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\,\mathrm{e7}\right]$

$A≔\left[\mathrm{e1}\,\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\,\mathrm{e7}\right]$

$\mathrm{Query}\left(A\,''Solvable''\right)$

$\mathrm{true}$

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

$\mathrm{false}$