A Lie algebra $\mathrm{\𝔤}\mathit{ }$is solvable if the $k$-th ideal ${\mathrm{\𝒟}}^k\left(g\right)$in the derived series for $\mathrm{\𝔤}\mathit{ }$ is 0 for some $k\ge 0$. Every nilpotent Lie algebra is solvable.

Query(Alg, "Solvable") returns true if Alg is a solvable Lie algebra and false otherwise. If the algebra is unspecified, then Query is applied to the current algebra.

Query(S, "Solvable") returns true if the subalgebra S is a solvable Lie algebra 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)\:$

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

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

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

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

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

$\mathrm{L3}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e3}\right]$

$\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)\:$

$\mathrm{Query}\left(\mathrm{Alg1}\,''Solvable''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(\mathrm{Alg2}\,''Solvable''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(\mathrm{Alg3}\,''Solvable''\right)$

$\mathrm{false}$

$\mathrm{Query}\left(\left[\mathrm{z1}\,\mathrm{z2}\right]\,''Solvable''\right)$

$\mathrm{true}$