The direct sum of two Lie algebras ${\mathrm{\𝔤}}_1$ and ${\mathrm{\𝔤}}_{2 }$is the vector space direct sum $\mathrm{\𝔤}\mathit{ }\mathit{\=}\mathit{ }{\mathrm{\𝔤}}_1 \oplus {\mathrm{\𝔤}}_2$ with Lie bracket

DirectSum(Summands, AlgName) creates a Lie algebra data structure for the direct sum of the Lie algebras listed in the first argument. The name given to the direct sum algebra is AlgName. The structure equations for the direct sum are displayed.

A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package.  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 command DirectSum is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form DirectSum(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-DirectSum(...).

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

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

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

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

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

$\mathrm{L3}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg3}\,\left[1\right]\right]\,\left[\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{L3}\,\left[z\right]\,\left[c\right]\right)\:$

$\mathrm{MultiplicationTable}\left(\mathrm{Alg1}\,''LieBracket''\right),\mathrm{MultiplicationTable}\left(\mathrm{Alg2}\,''LieBracket''\right),\mathrm{MultiplicationTable}\left(\mathrm{Alg3}\,''LieBracket''\right)$

$\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\right]\,\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=-2\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e3}\right]\,\left[\right]$

$\mathrm{L4}≔\mathrm{DirectSum}\left(\left[\mathrm{L1}\,\mathrm{L2}\right]\,\mathrm{Alg4}\right)$

$\mathrm{L4}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=-2\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e5}\right]$

$\mathrm{DGsetup}\left(\mathrm{L4}\right)\:$

$\mathrm{MultiplicationTable}\left(\mathrm{Alg4}\,''LieTable''\right)$

$\mathrm{L5}≔\mathrm{DirectSum}\left(\left[\mathrm{Alg1}\,\mathrm{Alg2}\,\mathrm{L3}\right]\,\mathrm{Alg5}\right)$

$\mathrm{L5}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=-2\mathrm{e4}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=\mathrm{e5}\right]$

$\mathrm{DGsetup}\left(\mathrm{L5}\right)\:$

$\mathrm{MultiplicationTable}\left(\mathrm{Alg5}\,''LieTable''\right)$