MultiplicationTable(LieAlgebraName, keyword) displays the form of structure equations for the Lie algebra or algebra dictated by the keyword.

If the keyword is "LieBracket", then the Lie brackets $\left[e_i\, e_j\right]$of the basis elements $\left\{e_1\, e_2\, ..\.\, e_n\right\}$are displayed in a two-dimensional array.

If the keyword is "AlgebraTable", then the non-commutative products $e_i\cdot e_j$of the basis elements $\left\{e_1\, e_2\, ..\.\, e_n\right\}$are displayed in a two-dimensional array.

If the keyword is "ExteriorDerivative", then the exterior derivatives $d\left({\mathrm{\θ}}^i\right)$of the dual basis elements $\left\{{\mathrm{\θ}}^1\, {\mathrm{\θ}}^2\, ..\. \, {\mathrm{\θ}}^n\right\}$are printed.

If the keyword is "LieDerivative", then the Lie derivatives ${\mathrm{\ℒ}}_{e_i}{\mathrm{\θ}}^j$ of the dual 1-forms $\left\{{\mathrm{\θ}}^1\, {\mathrm{\theta }}^2\, ..\. \, {\mathrm{\theta }}^n\right\}$ with respect to the basis vectors $\left\{e_1\, e_2\, ..\.\, e_n\right\}$are displayed in a two-dimensional array.

If LieAlgebraName is omitted, then the appropriate multiplication table of the current algebra is displayed.

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

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

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[5\right]\right]\,\left[\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,5\,3\right]\,1\right]\,\left[\left[4\,5\,4\right]\,1\right]\right]\right]\right)\:$

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

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

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

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

$d\left(\mathrm{\θ1}\right)\=-\mathrm{\θ2}\bigwedge \mathrm{\θ3}$

$d\left(\mathrm{\θ2}\right)\=0\mathrm{\θ1}\bigwedge \mathrm{\θ2}$

$d\left(\mathrm{\θ3}\right)\=-\mathrm{\θ2}\bigwedge \mathrm{\θ5}$

$d\left(\mathrm{\θ4}\right)\=-\mathrm{\θ4}\bigwedge \mathrm{\θ5}$

$d\left(\mathrm{\θ5}\right)\=0\mathrm{\θ1}\bigwedge \mathrm{\θ2}$

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

$\mathrm{L2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[4\right]\right]\,\left[\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,3\right]\,1\right]\,\left[\left[4\,2\,4\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[X\,Y\,U\,V\right]\,\left[\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\sigma }\,\mathrm{\tau }\right]\right)\:$

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

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

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

$d\left(\mathrm{\α}\right)\=-\mathrm{\β}\bigwedge \mathrm{\σ}$

$d\left(\mathrm{\β}\right)\=0\mathrm{\α}\bigwedge \mathrm{\β}$

$d\left(\mathrm{\σ}\right)\=-\mathrm{\β}\bigwedge \mathrm{\τ}$

$d\left(\mathrm{\τ}\right)\=-\mathrm{\β}\bigwedge \mathrm{\τ}$

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

$\mathrm{L3}≔\mathrm{AlgebraLibraryData}\left(''Quaternions''\,H\right)$

$\mathrm{L3}:=\left[{\mathrm{e1}}^2\=\mathrm{e1}\,\mathrm{e1}\.\mathrm{e2}\=\mathrm{e2}\,\mathrm{e1}\.\mathrm{e3}\=\mathrm{e3}\,\mathrm{e1}\.\mathrm{e4}\=\mathrm{e4}\,\mathrm{e2}\.\mathrm{e1}\=\mathrm{e2}\,{\mathrm{e2}}^2\=-\mathrm{e1}\,\mathrm{e2}\.\mathrm{e3}\=\mathrm{e4}\,\mathrm{e2}\.\mathrm{e4}\=-\mathrm{e3}\,\mathrm{e3}\.\mathrm{e1}\=\mathrm{e3}\,\mathrm{e3}\.\mathrm{e2}\=-\mathrm{e4}\,{\mathrm{e3}}^2\=-\mathrm{e1}\,\mathrm{e3}\.\mathrm{e4}\=\mathrm{e2}\,\mathrm{e4}\.\mathrm{e1}\=\mathrm{e4}\,\mathrm{e4}\.\mathrm{e2}\=\mathrm{e3}\,\mathrm{e4}\.\mathrm{e3}\=-\mathrm{e2}\,{\mathrm{e4}}^2\=-\mathrm{e1}\right]$

$\mathrm{DGsetup}\left(\mathrm{L3}\,\left[e\,i\,j\,k\right]\,\left[\mathrm{\theta }\right]\right)$

$\mathrm{algebra\; name:\; H}$

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