Let $g$and $k$$$be Lie algebras and let $\mathrm{\φ}\:g \to k$be a Lie algebra homomorphism .The kernel of $\mathrm{\φ}$is the ideal of vectors ker($\mathrm{\φ}\) \= \{$$x \in g$| $\mathrm{\φ}\left(x\right) \= 0\}\.$The image of $\mathrm{\φ}$is the subalgebra of vectors im$\left(\mathrm{\φ}\right)$ = $\{y \in k \| y \= \mathrm{\φ}\left(x\right)$ for some $x \in g\}$. If $S$ is a subalgebra of $k$, then the inverse image of $S$with respect to $\mathrm{\φ}$is the subalgebra  ${\mathrm{\φ}}^{-1}\left(S\right) \= \{x \in g \| \mathrm{\φ}\left(x\right) \in S\}$.

HomomorphismSubalgebras($\mathbf{\φ}$, "Kernel") calculates ker$\left(\mathrm{\φ}\right)\.$  A list of independent vectors defining a basis for the kernel is returned.  If ker($\mathrm{\φ}\) \= 0\,$then an empty list is returned.

HomomorphismSubalgebras($\mathbf{\φ}$, "Image") calculates im($\mathrm{\φ}$).  A list of independent vectors defining a basis for the image is returned. If  im($\mathrm{\φ}\) \= 0\,$then an empty list is returned.

HomomorphismSubalgebras($\mathbf{\φ}$, S, "InverseImage") calculates ${\mathrm{\φ}}^{-1}\left(S\right)$. A list of independent vectors defining a basis for the inverse image is returned. If ${\mathrm{\phi }}^{-1}\left(S\right)$ = 0, then an empty list is returned.

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

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

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

$\mathrm{DGsetup}\left(\mathrm{L1}\,\left[x\right]\,\left[\mathrm{\alpha }\right]\right)\:$

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg2}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,2\right]\,1\right]\,\left[\left[3\,4\,3\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[y\right]\,\left[\mathrm{\beta }\right]\right)\:$

$\mathrm{print}\left(\mathrm{MultiplicationTable}\left(\mathrm{Alg1}\,''LieBracket''\right)\,\mathrm{MultiplicationTable}\left(\mathrm{Alg2}\,''LieBracket''\right)\right)$

$\left[\left[\mathrm{x2}\,\mathrm{x3}\right]\=\mathrm{x1}\right]\,\left[\left[\mathrm{y1}\,\mathrm{y4}\right]\=\mathrm{y2}\,\left[\mathrm{y3}\,\mathrm{y4}\right]\=\mathrm{y3}\right]$

$\mathrm{\Phi }≔\mathrm{Transformation}\left(\left[\left[\mathrm{x1}\,0\&mult\mathrm{y1}\right]\,\left[\mathrm{x2}\,\mathrm{y2}\right]\,\left[\mathrm{x3}\,\mathrm{y3}\right]\right]\right)$

$\mathrm{\Φ}≔\left[\left[\mathrm{x1}\,0\mathrm{y1}\right]\,\left[\mathrm{x2}\,\mathrm{y2}\right]\,\left[\mathrm{x3}\,\mathrm{y3}\right]\right]$

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

$\mathrm{true}$

$\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Phi }\,''Kernel''\right)$

$\left[\mathrm{x1}\right]$

$\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Phi }\,''Image''\right)$

$\left[\mathrm{y2}\,\mathrm{y3}\right]$

$\mathrm{S1}≔\left[\mathrm{y3}\,\mathrm{y4}\right]\:$

$\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Phi }\,\mathrm{S1}\,''InverseImage''\right)$

$\left[\mathrm{x1}\,-\mathrm{x3}\right]$