A matrix $A$ defines a Lie algebra homomorphism from a Lie algebra $\mathrm{\𝔤}$ to a Lie algebra $\mathrm{\𝔥}$ if the linear transformation ${L_A}_{}$ satisfies $L_A\left(\left[x\, y\right]\right) \= \left[L_A\left(x\right)\, L_A\left(y\right)\right]$ for all $x\, y \in \mathrm{\𝔤}$.

Query(Alg1, Alg2, A, "Homomorphism") returns true if the matrix A defines a Lie algebra homomorphism from $\mathrm{\𝔤}$ to $\mathrm{\𝔥}$ and false otherwise.

Query(Alg1, Alg2, phi, "Homomorphism") returns true if the transformation $\mathrm{\φ}$ defines a Lie algebra homomorphism $\mathrm{\φ}\: \mathrm{\𝔤}$ -> $\mathrm{\𝔥}$ and false otherwise.

Query(Alg1, Alg2, parm, "Homomorphism") 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 A is a homomorphism; Eq is the defining set of equations for the parameters parm in order that the matrix A be a homomorphism; 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)\:$

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

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

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

$\mathrm{A1}≔\mathrm{AdjointExp}\left(r\mathrm{e1}+s\mathrm{e2}+t\mathrm{e3}\right)$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg1}\,\mathrm{A1}\,''Homomorphism''\right)$

$\mathrm{true}$

$\mathrm{Outer}≔\mathrm{Derivations}\left(''Outer''\right)$

$\mathrm{A2}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixExponential}\left(t\mathrm{Outer}\left[1\right]\right)$

$T≔\mathrm{Transformation}\left(\mathrm{Alg1}\,\mathrm{Alg1}\,\mathrm{A2}\right)$

$T:=\left[\left[\mathrm{e1}\,{\ⅇ}^t\mathrm{e1}\right]\,\left[\mathrm{e2}\,\mathrm{e2}\right]\,\left[\mathrm{e3}\,{\ⅇ}^t\mathrm{e3}\right]\,\left[\mathrm{e4}\,\mathrm{e4}\right]\right]$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg1}\,T\,''Homomorphism''\right)$

$\mathrm{true}$

$\mathrm{L2}≔\mathrm{QuotientAlgebra}\left(\left[\mathrm{e1}\right]\,\left[\mathrm{e2}\,\mathrm{e3}\,\mathrm{e4}\right]\,\mathrm{Alg2}\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[x\right]\,\left[\mathrm{\alpha }\right]\right)\:$

$\left[\begin{array}{ccccc}& \mathrm{`|`}& {{\mathrm{x1}}_{}}_{}& {{\mathrm{x2}}_{}}_{}& {{\mathrm{x3}}_{}}_{}\\ & \mathrm{---}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ {{\mathrm{x1}}_{}}_{}& \mathrm{`|`}& 0& 0& {\mathrm{x1}}_{}\\ {{\mathrm{x2}}_{}}_{}& \mathrm{`|`}& 0& 0& 0\\ {{\mathrm{x3}}_{}}_{}& \mathrm{`|`}& -{\mathrm{x1}}_{}& 0& 0\end{array}\right]$

$\mathrm{A3}≔\mathrm{Matrix}\left(\left[\left[0\,1\,0\,0\right]\,\left[0\,0\,1\,0\right]\,\left[0\,0\,0\,1\right]\right]\right)$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,\mathrm{A3}\,''Homomorphism''\right)$

$\mathrm{true}$

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg3}\,\left[2\right]\right]\,\left[\left[\left[1\,2\,1\right]\,1\right]\right]\right]\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\right)\:$

$\mathrm{A4}≔\mathrm{Matrix}\left(\left[\left[\mathrm{a1}\,\mathrm{a2}\right]\,\left[\mathrm{a3}\,\mathrm{a4}\right]\,\left[\mathrm{a5}\,\mathrm{a6}\right]\,\left[\mathrm{a7}\,\mathrm{a8}\right]\right]\right)$

$\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},B≔\mathrm{Query}\left(\mathrm{Alg3}\,\mathrm{Alg1}\,\mathrm{A4}\,\left\{\mathrm{a1}\,\mathrm{a2}\,\mathrm{a3}\,\mathrm{a4}\,\mathrm{a5}\,\mathrm{a6}\,\mathrm{a7}\,\mathrm{a8}\right\}\,''Homomorphism''\right)$

$\mathrm{EQ}$

$\left\{0\,\mathrm{a5}\,\mathrm{a7}\,-\mathrm{a5}\,-\mathrm{a7}\,-\mathrm{a3}\mathrm{a8}\+\mathrm{a4}\mathrm{a7}\+\mathrm{a3}\,\mathrm{a3}\mathrm{a8}-\mathrm{a4}\mathrm{a7}-\mathrm{a3}\,-\mathrm{a1}\mathrm{a8}\+\mathrm{a2}\mathrm{a7}-\mathrm{a3}\mathrm{a6}\+\mathrm{a4}\mathrm{a5}\+\mathrm{a1}\,\mathrm{a1}\mathrm{a8}-\mathrm{a2}\mathrm{a7}\+\mathrm{a3}\mathrm{a6}-\mathrm{a4}\mathrm{a5}-\mathrm{a1}\right\}$

$B$