Let $G$be a Lie group with multiplication $\*$ and define diffeomorphisms $L_a\:G\to G$ and $R_a\:G \to G$by $L_a\left(x\right) \= a\*g$ and $R_a\left(g\right) \= x\*a$. A vector field $X$ on $G$ is left invariant if ${\left(L_a\right)}_{\*}\left(X_b\right) \= X_{\mathrm{ab} }$and right invariant if ${\left(R_a\right)}_{\*}\left(X_b\right) \= X_{\mathrm{ba} }$for all $b \in G$. A differential form $\mathrm{\ω}$ on $G$ is left invariant if $L_a^{\*}\left({\mathrm{\ω}}_{\mathrm{ab}}\right)$ = ${\mathrm{\ω}}_b$and right invariant if $R_a^{\*}\left({\mathrm{\ω}}_{\mathrm{ba}}\right)$ = ${\mathrm{\ω}}_b$. Every Lie group admits a set of $n\=$dim$\left(G\right)$pointwise linearly independent left or right invariant vector fields (an invariant frame) and a set of $n\=$dim$\left(G\right)$pointwise linearly independent left or right invariant 1-forms (an invariant coframe). Indeed, the infinitesimal generators for the group action defined by $R_a$give a left invariant frame while the infinitesimal generators for the group action defined by $L_a$give a right invariant frame.

The command InvariantVectorsAndForms(LG) returns up to a sequence of four lists XL, OmegaL, XR, OmegaR, where XL is a frame of left invariant vector fields, OmegaL is a coframe of left invariant 1-forms, XR is a frame of right invariant vector fields, and OmegaR is a frame of right invariant 1-forms.

The output option allows the user to dictate precisely which lists of invariant vector fields and forms are returned and the order in which they are returned. The default is output = ["LeftVectors", "LeftForms", "RightVectors", "RightForms"].

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

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

$M≔\mathrm{Matrix}\left(\left[\left[x\,y\,z\right]\,\left[0\,w\,0\right]\,\left[0\,0\,1\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,w\right]\,G\right)$

$\mathrm{frame\; name:\; G}$

$\mathrm{LG}≔\mathrm{LieGroup}\left(M\,G\right)$

$\mathrm{LG}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{XL},\mathrm{OmegaL}≔\mathrm{InvariantVectorsAndForms}\left(\mathrm{LG}\,\mathrm{output}=\left[''LeftVectors''\,''LeftForms''\right]\right)$

$\mathrm{XL}\,\mathrm{OmegaL}:=\left[x\mathrm{D\_x}\,x\mathrm{D\_y}\,x\mathrm{D\_z}\,y\mathrm{D\_y}\+w\mathrm{D\_w}\right]\,\left[\frac{\mathrm{dx}}{x}\,\frac{\mathrm{dy}}{x}-\frac{y\mathrm{dw}}{xw}\,\frac{\mathrm{dz}}{x}\,\frac{\mathrm{dw}}{w}\right]$

$\mathrm{XR}≔\mathrm{InvariantVectorsAndForms}\left(\mathrm{LG}\,\mathrm{output}=\left[''RightVectors''\right]\right)$

$\mathrm{XR}:=\left[x\mathrm{D\_x}\+y\mathrm{D\_y}\+z\mathrm{D\_z}\,w\mathrm{D\_y}\,\mathrm{D\_z}\,w\mathrm{D\_w}\right]$