There are various equivalent definitions for a semisimple Lie algebra. One definition, called Cartan's criterion, states that a Lie algebra is semisimple if its Killing form is non-degenerate. This is the test applied by Query.

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

Query(S, "Semisimple") returns true if the subalgebra S is a semisimple 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)\:$

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

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

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

$\mathrm{L2}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e3}\right]$

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

$\mathrm{L3}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=-\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=-\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=-\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=2\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=-\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=2\mathrm{e5}\right]$

$\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)\:$

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

$\mathrm{false}$

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

$\mathrm{true}$

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

$\mathrm{false}$

$S≔\left[\mathrm{z3}\,\mathrm{z4}\,\mathrm{z5}\right]\:$

$\mathrm{Query}\left(S\,''Semisimple''\right)$

$\mathrm{true}$