A pair of subalgebras $R\, S$ in a Lie algebra $\mathrm{\𝔤}\mathit{ }$define a Levi decomposition if $R$is the radical of $\mathrm{\𝔤}$, $S$ is a semi-simple subalgebra, and $g \= R \oplus S$(vector space direct sum). Since the radical is an ideal we have $\left[R\,\.R\right] \subset R$, $\left[R\, S\right] \subset R\,$and $\left[S\, S\right] \subset S\.$The radical $R$ is unique, the semisimple subalgebra $S$in a Levi decomposition is not.

Query([R, S], "LeviDecomposition") returns true if the pair R, S is a Levi decomposition of $\mathrm{\𝔤}\mathit{ }$ and false otherwise.

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

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

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

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg2}\,\left[3\right]\right]\,\left[\left[\left[1\,2\,1\right]\,1\right]\,\left[\left[1\,3\,2\right]\,-2\right]\,\left[\left[2\,3\,3\right]\,1\right]\right]\right]\right)\:$

$\mathrm{L3}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg3}\,\left[5\right]\right]\,\left[\left[\left[1\,3\,2\right]\,-1\right]\,\left[\left[1\,4\,1\right]\,-1\right]\,\left[\left[2\,4\,2\right]\,1\right]\,\left[\left[2\,5\,1\right]\,-1\right]\,\left[\left[3\,4\,3\right]\,2\right]\,\left[\left[3\,5\,4\right]\,-1\right]\,\left[\left[4\,5\,5\right]\,2\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L1}\,\left[x\right]\,\left[a\right]\right)\:$$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[y\right]\,\left[b\right]\right)\:$$\mathrm{DGsetup}\left(\mathrm{L3}\,\left[z\right]\,\left[c\right]\right)\:$

$\mathrm{print}\left(\mathrm{MultiplicationTable}\left(\mathrm{Alg1}\,''LieTable''\right)\,\mathrm{MultiplicationTable}\left(\mathrm{Alg2}\,''LieTable''\right)\,\mathrm{MultiplicationTable}\left(\mathrm{Alg3}\,''LieTable''\right)\right)$

$\mathrm{R1}≔\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\right]\:$$\mathrm{S1}≔\left[\right]\:$

$\mathrm{Query}\left(\left[\mathrm{R1}\,\mathrm{S1}\right]\,''LeviDecomposition''\right)$

$\mathrm{true}$

$\mathrm{R2}≔\left[\right]\:$$\mathrm{S2}≔\left[\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\right]\:$

$\mathrm{Query}\left(\left[\mathrm{R2}\,\mathrm{S2}\right]\,''LeviDecomposition''\right)$

$\mathrm{true}$

$\mathrm{R3}≔\left[\mathrm{z1}\,\mathrm{z2}\right]\:$$\mathrm{S3}≔\left[\mathrm{z3}\,\mathrm{z4}\,\mathrm{z5}\right]\:$

$\mathrm{Query}\left(\left[\mathrm{R3}\,\mathrm{S3}\right]\,''LeviDecomposition''\right)$

$\mathrm{true}$

$\mathrm{SS0}≔\mathrm{ComplementaryBasis}\left(\mathrm{R3}\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\,\mathrm{z5}\right]\,k\right)$

$\mathrm{SS0}:=\left[\mathrm{k1}\mathrm{z1}\+\mathrm{k2}\mathrm{z2}\+\mathrm{z3}\,\mathrm{k3}\mathrm{z1}\+\mathrm{k4}\mathrm{z2}\+\mathrm{z4}\,\mathrm{k5}\mathrm{z1}\+\mathrm{k6}\mathrm{z2}\+\mathrm{z5}\right]\,\left\{\mathrm{k1}\,\mathrm{k2}\,\mathrm{k3}\,\mathrm{k4}\,\mathrm{k5}\,\mathrm{k6}\right\}$

$\mathrm{TF},\mathrm{Eq},\mathrm{SOL},\mathrm{SubAlgList}≔\mathrm{Query}\left(\mathrm{SS0}\,''Subalgebra''\right)$

$\mathrm{TF}\,\mathrm{Eq}\,\mathrm{SOL}\,\mathrm{SubAlgList}:=\mathrm{true}\,\left\{0\,-3\mathrm{k1}\,-3\mathrm{k6}\,-\mathrm{k2}\+\mathrm{k3}\,\mathrm{k4}\+\mathrm{k5}\,-\mathrm{k5}-\mathrm{k4}\right\}\,\left[\left\{\mathrm{k1}\=0\,\mathrm{k2}\=\mathrm{k3}\,\mathrm{k3}\=\mathrm{k3}\,\mathrm{k4}\=-\mathrm{k5}\,\mathrm{k5}\=\mathrm{k5}\,\mathrm{k6}\=0\right\}\right]\,\left[\left[\mathrm{k3}\mathrm{z2}\+\mathrm{z3}\,\mathrm{k3}\mathrm{z1}-\mathrm{k5}\mathrm{z2}\+\mathrm{z4}\,\mathrm{k5}\mathrm{z1}\+\mathrm{z5}\right]\right]$

$\mathrm{S4}≔\mathrm{SubAlgList}\left[1\right]$

$\mathrm{S4}:=\left[\mathrm{k3}\mathrm{z2}\+\mathrm{z3}\,\mathrm{k3}\mathrm{z1}-\mathrm{k5}\mathrm{z2}\+\mathrm{z4}\,\mathrm{k5}\mathrm{z1}\+\mathrm{z5}\right]$

$\mathrm{Query}\left(\left[\mathrm{R3}\,\mathrm{S4}\right]\,''LeviDecomposition''\right)$

$\mathrm{true}$