A matrix $A$ is a derivation for a Lie algebra $\mathrm{\𝔤}\mathit{ }$ if the associated linear transformation mapping $L_A\: \mathrm{\𝔤} \to \mathrm{\𝔤}$  satisfies  $L_A$($\left[x\, y\right]\) \= \left[L_A\left(x\right)\, y\right] \+\left[x\, L_A\left(y\right)\right]$ for all $x\, y \in \mathrm{\𝔤}$.

Query(Alg, A, "Derivation") returns true if the matrix A or transformation defines a derivation for the Lie algebra g and false otherwise.

Query(Alg, A, parm, "Derivation") returns a 4-tuple TF, Eq, Soln, B.  Here TF is true if Maple finds a set of values for the parameters for which the Matrix or transformation A is a derivation; Eq is the defining set of equations for the parameters parm in order that the matrix A be a derivation; Soln is a list of solutions to the equations Eq; and B is the list of Matrices obtained by evaluating A on the solutions in the list Soln.

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)\:$

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,1\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,1\right]\,\left[\left[3\,4\,1\right]\,0\right]\right]\right]\right)$

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

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

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

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

$A≔\mathrm{Adjoint}\left(\mathrm{e1}-2\mathrm{e3}+\mathrm{e4}\right)$

$A:=\left[\begin{array}{rrrr}-1& 2& 0& 1\\ 0& -1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

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

$\mathrm{true}$

$A≔\mathrm{Matrix}\left(\left[\left[0\,0\,\mathrm{a1}\,\mathrm{a2}\right]\,\left[0\,0\,0\,\mathrm{a3}\right]\,\left[0\,0\,0\,0\right]\,\left[0\,0\,0\,0\right]\right]\right)$

$A:=\left[\begin{array}{cccc}0& 0& \mathrm{a1}& \mathrm{a2}\\ 0& 0& 0& \mathrm{a3}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

$T≔\mathrm{Transformation}\left(\mathrm{Alg1}\,\mathrm{Alg1}\,A\right)$

$T≔\left[\left[\mathrm{e1}\,0\mathrm{e1}\right]\,\left[\mathrm{e2}\,0\mathrm{e1}\right]\,\left[\mathrm{e3}\,\mathrm{a1}\mathrm{e1}\right]\,\left[\mathrm{e4}\,\mathrm{a2}\mathrm{e1}\+\mathrm{a3}\mathrm{e2}\right]\right]$

$\mathrm{TF},\mathrm{Eq},\mathrm{Soln},\mathrm{Der}≔\mathrm{Query}\left(\mathrm{Alg1}\,T\,\left\{\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\right\}\,''Derivation''\right)$

$\mathrm{TF}\,\mathrm{Eq}\,\mathrm{Soln}\,\mathrm{Der}≔\mathrm{true}\,\left\{0\,-\mathrm{a3}\+\mathrm{a1}\right\}\,\left[\left\{\mathrm{a1}\=\mathrm{a3}\,\mathrm{a2}\=\mathrm{a2}\,\mathrm{a3}\=\mathrm{a3}\right\}\right]\,\left[\left[\left[\mathrm{e1}\,0\mathrm{e1}\right]\,\left[\mathrm{e2}\,0\mathrm{e1}\right]\,\left[\mathrm{e3}\,\mathrm{a3}\mathrm{e1}\right]\,\left[\mathrm{e4}\,\mathrm{a2}\mathrm{e1}\+\mathrm{a3}\mathrm{e2}\right]\right]\right]$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Der}\left[1\right]\,''Derivation''\right)$

$\mathrm{true}$