A collection of subalgebras $S_1\, S_2$... of a Lie algebra $\mathrm{\𝔤}\mathit{ }$define a direct sum decomposition of $\mathrm{\𝔤}\mathit{ }\mathit{ }$if  $\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }{S}_1\oplus S_{2 }\oplus \cdot \cdot \cdot$ (vector space direct sum) and  $\left[S_i\, S_j\right] \=0$for $i \ne j$.

Query([S1, S2, ... ], "DirectSumDecomposition") returns true if the subspaces  $S_1\, S_2\, ..\.$ define a direct sum decomposition of the Lie algebra $\mathrm{\𝔤}$$$and false otherwise

Query(B, [d1, d2, ... ], "DirectSumDecomposition") returns true if the first ${d_1}_{}$vectors in $B$, the second $d_{2 }$vectors in $B$, ... define a direct sum 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)\:$

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

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

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

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

$\mathrm{S1}≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\:$$\mathrm{S2}≔\left[\mathrm{e4}\,\mathrm{e5}\right]\:$$\mathrm{S3}≔\left[\mathrm{e6}\right]\:$

$\mathrm{Query}\left(\left[\mathrm{S1}\,\mathrm{S2}\,\mathrm{S3}\right]\,''DirectSumDecomposition''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(\left[\mathrm{S1}\,\mathrm{S2}\right]\,''DirectSumDecomposition''\right)$

$\mathrm{false}$

$B≔\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\,\mathrm{e5}\,\mathrm{e6}\right]\:$

$\mathrm{Query}\left(B\,\left[3\,3\right]\,''DirectSumDecomposition''\right)$

$\mathrm{true}$

$\mathrm{Query}\left(B\,\left[2\,2\,2\right]\,''DirectSumDecomposition''\right)$

$\mathrm{false}$