In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.  The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.

An important class of Lie algebras is given by the matrix Lie algebras, where the Lie algebra bracket is given by the matrix commutator $\left[a\,b\right] \= a\cdot b - b\cdot a$  The program LieAlgebraData(MatrixAlgebra, AlgName) will create the Lie algebra data structure for the matrix algebra whose basis is the given list of matrices MatrixAlgebra.

If the given matrices are complex matrices which form a real Lie algebra, then the keyword argument coefficienttype = "real" will ensure that real structure constants are calculated.

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

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

$M≔\mathrm{map}\left(\mathrm{Matrix}\,\left[\left[\left[-1\,2\,-1\right]\,\left[0\,1\,0\right]\,\left[2\,-2\,2\right]\right]\,\left[\left[1\,0\,0\right]\,\left[2\,-1\,1\right]\,\left[2\,-2\,2\right]\right]\,\left[\left[0\,0\,0\right]\,\left[1\,-1\,1\right]\,\left[2\,-2\,2\right]\right]\,\left[\left[-1\,2\,-1\right]\,\left[1\,0\,0\right]\,\left[4\,-4\,2\right]\right]\,\left[\left[4\,-4\,2\right]\,\left[2\,-2\,1\right]\,\left[-2\,2\,-1\right]\right]\right]\right)$

$P≔M\left[1\right]·M\left[5\right]-M\left[5\right]·M\left[1\right]$

$\mathrm{GetComponents}\left(P\,M\right)$

$\left[-1\,1\,0\,0\,0\right]$

$Q≔-M\left[1\right]+M\left[2\right]$

$L≔\mathrm{LieAlgebraData}\left(M\,\mathrm{Ex1}\right)$

$L:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e2}-\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e4}\+\mathrm{e3}-\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e5}\right]\=\mathrm{e2}-\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e4}\+\mathrm{e3}-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=2\mathrm{e2}-2\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e4}\+\mathrm{e3}-\mathrm{e1}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e4}\+\mathrm{e3}\+\mathrm{e2}-2\mathrm{e1}\right]$

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

$\mathrm{Query}\left(''Indecomposable''\right)$

$\mathrm{false}$

$M≔\mathrm{map}\left(\mathrm{Matrix}\,\left[\left[\left[1\,0\right]\,\left[0\,-1\right]\right]\,\left[\left[0\,1\right]\,\left[0\,0\right]\right]\,\left[\left[0\,0\right]\,\left[1\,0\right]\right]\,\left[\left[\mathrm{I}\,0\right]\,\left[0\,-\mathrm{I}\right]\right]\,\left[\left[0\,\mathrm{I}\right]\,\left[0\,0\right]\right]\,\left[\left[0\,0\right]\,\left[\mathrm{I}\,0\right]\right]\right]\right)$

$\mathrm{LieAlgebraData}\left(M\,\mathrm{alg2}\,\mathrm{coefficienttype}=''real''\right)$

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