A collection of subspaces  $f_0\, f_1\, ..\.\, f_N$  of a Lie algebra $\mathrm{\𝔤}$ defines a decreasing filtration of $\mathrm{\𝔤}\mathit{ }$ if  [i] $$$f_i \subset f_j$  for $i\ge j$, [ii] $\left[f_i\, f_j\right] \subset f_{i\+j }$ for  $i \+j \le N\,$and [iii] $\left[f_i\, f_j\right] \= 0$for $i \+j \>N$.

Query([f0, f1, ... fN], "Filtration") returns true if the subspaces  $f_0\, f_1\, ..\.\, f_N$ define a decreasing filtration of the Lie algebra $\mathrm{\𝔤}\mathit{ }$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)\:$

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

$\mathrm{L1}:=\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+\mathrm{e3}\right]$

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

$\mathrm{f0}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\:$$\mathrm{f1}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\:$$\mathrm{f2}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$$\mathrm{f3}≔\left[\mathrm{e1}\right]\:$$\mathrm{f4}≔\left[\right]\:$

$\mathrm{Query}\left(\left[\mathrm{f0}\,\mathrm{f1}\,\mathrm{f2}\,\mathrm{f3}\,\mathrm{f4}\right]\,''Filtration''\right)$

$\mathrm{true}$

$\mathrm{f0}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\:$$\mathrm{f1}≔\left[\mathrm{e1}\,\mathrm{e2}\right]\:$$\mathrm{f2}≔\left[\mathrm{e2}\right]\:$$\mathrm{f3}≔\left[\right]\:$

$\mathrm{infolevel}\left[\mathrm{Query}\right]≔2\:$

$\mathrm{Query}\left(\left[\mathrm{f0}\,\mathrm{f1}\,\mathrm{f2}\,\mathrm{f3}\,\mathrm{f4}\right]\,''Filtration''\right)$

bracket of subspaces with weights 0 and 0 is [2*e1, e2, e2+e3]

bracket of subspaces with weights 0 and 1 is [-e1, -e2]
bracket of subspaces with weights 0 and 2 is [-e1, -e2]

$\mathrm{false}$