The command DGsetup is used to initialize a Lie algebra, that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory. The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.

One commonly used format for the structure equations of a Lie algebra $\mathrm{\𝔤}$ is the set of exterior derivative equations for the dual 1-forms of the Lie algebra.  For a 1-form $\mathrm{\θ}$ in the dual of a Lie algebra, the exterior derivative is the 2-form defined by $d\left(\mathrm{\θ}\right)\left(x\,y\right) \= -{\mathrm{\θ}}_{}\left(x\,y\right)$, for $x\,y \in \mathrm{\𝔤}$. The function LieAlgebraData enables one to create a Lie algebra in Maple from a list of exterior derivative equations.

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

with(DifferentialGeometry): with(LieAlgebras):

FormStrEq := [d(t1) = -t2 &w t3, d(t2) = - t1 &w t3, d(t3) = a*t1 &w t2 + b*t2 &w t3 + c*t1 &w t3];

$\mathrm{FormStrEq}:=\left[d\left(\mathrm{t1}\right)\=-\mathrm{t2}\&w\mathrm{t3}\,d\left(\mathrm{t2}\right)\=-\mathrm{t1}\&w\mathrm{t3}\,d\left(\mathrm{t3}\right)\=a\mathrm{t1}\&w\mathrm{t2}\+b\mathrm{t2}\&w\mathrm{t3}\+c\mathrm{t1}\&w\mathrm{t3}\right]$

Basis := [t1, t2, t3];

$\mathrm{Basis}:=\left[\mathrm{t1}\,\mathrm{t2}\,\mathrm{t3}\right]$

L := LieAlgebraData(FormStrEq, Basis, Ex1);

$L:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-a\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e2}-c\mathrm{e3}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}-b\mathrm{e3}\right]$

DGsetup(L);

$\mathrm{Lie\; algebra:\; Ex1}$

MultiplicationTable("ExteriorDerivative");

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

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

$d\left(\mathrm{\θ3}\right)\=a\mathrm{\θ1}\bigwedge \mathrm{\θ2}\+c\mathrm{\θ1}\bigwedge \mathrm{\θ3}\+b\mathrm{\θ2}\bigwedge \mathrm{\θ3}$

TF, EQ, SOLN, LIEALG := Query({a, b, c}, "Jacobi");

$\mathrm{TF}\,\mathrm{EQ}\,\mathrm{SOLN}\,\mathrm{LIEALG}:=\mathrm{true}\,\left\{0\,b\,-c\right\}\,\left[\left\{a\=a\,b\=0\,c\=0\right\}\right]\,\left[\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-a\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\right]\right]$