In the LieAlgebras package, 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 describing the structure equations of a Lie algebra is to specify the structure constants $C$as an $n \times n \times n$array, where $n$ is the dimension of the Lie algebra to be created. The structure constants $C_{\mathrm{ijk}}$are defined to be the Lie brackets of the basis elements of the Lie algebra by $\left[e_i\, e_k\right] \=\sum _{k\=1}^n{C}_{\mathrm{ijk}} e_k$.

The function LieAlgebraData enables one to create a Lie algebra in Maple from an array $C$.  Only the entries $C_{\mathrm{ijk}}$$$, where $i<j\&comma;$need be specified.

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):

C := Array(1..4, 1..4, 1..4, 0):

C[1, 4, 1] := a:

C[2, 3, 1] := b:

C[3, 4, 2] := c:

C[3, 4, 3] := a:

L1 := LieAlgebraData(C, Ex1);

$\mathrm{L1}:=\left[\left[\mathrm{e1}\&comma;\mathrm{e4}\right]\&equals;a\mathrm{e1}\&comma;\left[\mathrm{e2}\&comma;\mathrm{e3}\right]\&equals;b\mathrm{e1}\&comma;\left[\mathrm{e3}\&comma;\mathrm{e4}\right]\&equals;c\mathrm{e2}\&plus;a\mathrm{e3}\right]$

DGsetup(L1):

Query("Jacobi");

$\mathrm{true}$

K := Array(1..4, 1..4, 1..4, (i, j ,k) -> Tools:-DGinfo([i, j, k], "LieBracketStructureFunction"));

K[1, 4, 1];

$a$

K[4, 1, 1];

$-a$