MatrixSubalgebra - Maple Help

LieAlgebras[MatrixSubalgebra] - find the subalgebra of a Lie algebra which preserves a collection of tensors or subspaces of tensors

Calling Sequences

MatrixSubalgebra(rho, Inv)

MatrixSubalgebra($\mathrm{alg}$, M, Inv)

MatrixSubalgebra($\mathrm{alg}$, Gamma, Inv)

MatrixSubalgebra($\mathrm{alg}$1, Inv)

Parameters

rho     - a representation of a Lie algebra

Inv     - a list, where each element is a tensor or a list of tensors

alg     - a name or a string, the name of an initialized Lie algebra 

M       - a list of square matrices defining a Lie algebra, with the same structure equations as 

Gamma   - a list of vector fields defining a Lie algebra, with the same structure equations as 

alg1    - a name or a string, the name of an initialized Lie algebra which has been created by the command $\mathrm{SimpleLieAlgebraData}$

Description

 • Let  be a vector space and  a linear transformation (not necessarily invertible).  Let  be a type (1,1) tensor on Then the (1,1) tensor   is defined by

where  and *.

If ${{\mathrm{φ}}^{i}}_{j}^{}$ and  are the components of and  with respect to a basis for (and dual basis for ${V}^{*}$), then  .

 • This formula extends in the natural way to define  for any tensor One says that  is $\mathrm{φ}-$invariant if
 • Let g be a Lie algebra and let  be a representation of The set a is a subalgebra of g.  Likewise, if T is a subspace of tensors, then the set b for all   is also a subalgebra of g.
 • The command  allows one to make subalgebras via this general construction.  The argument Inv  is a list where each element is a tensor or a list of  tensors. For example, if  then  calculates the subalgebra consisting of   such that  span span
 • When a Lie algebra is created with the command SimpleLieAlgebraData, its standard matrix representation is encoded in the Lie algebra data structure for that algebra. For such algebras, the construction of subalgebras via invariant tensors can be performed without explicitly specifying a representation.



Examples

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

Example 1.

We construct the Lie algebras $\mathrm{so}\left(3\right)$ and  $\mathrm{so}$(3)⊕$\mathrm{so}$(2)  as subalgebras of $\mathrm{so}$(5). First, here are the 5×5 skew-symmetric matrices which define $\mathrm{so}\left(5\right).$

 > $A≔\mathrm{map}\left(\mathrm{Matrix},\left[\left[\left[0,-1,0,0,0\right],\left[1,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right]\right],\left[\left[0,0,-1,0,0\right],\left[0,0,0,0,0\right],\left[1,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right]\right],\left[\left[0,0,0,-1,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[1,0,0,0,0\right],\left[0,0,0,0,0\right]\right],\left[\left[0,0,0,0,-1\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[1,0,0,0,0\right]\right],\left[\left[0,0,0,0,0\right],\left[0,0,-1,0,0\right],\left[0,1,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right]\right],\left[\left[0,0,0,0,0\right],\left[0,0,0,-1,0\right],\left[0,0,0,0,0\right],\left[0,1,0,0,0\right],\left[0,0,0,0,0\right]\right],\left[\left[0,0,0,0,0\right],\left[0,0,0,0,-1\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,1,0,0,0\right]\right],\left[\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,-1,0\right],\left[0,0,1,0,0\right],\left[0,0,0,0,0\right]\right],\left[\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,-1\right],\left[0,0,0,0,0\right],\left[0,0,1,0,0\right]\right],\left[\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,0\right],\left[0,0,0,0,-1\right],\left[0,0,0,1,0\right]\right]\right]\right)$

Calculate the structure equations and initialize.

 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(A,\mathrm{so5}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}\right]$ (2.1)
 > $\mathrm{lprint}\left(%\right)$
 _DG([["LieAlgebra", so5, [10, table( [ ] )]], [[[1, 2, 5], 1], [[1, 3, 6], 1], [[1, 4, 7], 1], [[1, 5, 2], -1], [[1, 6, 3], -1], [[1, 7, 4], -1], [[2, 3, 8], 1], [[2, 4, 9], 1], [[2, 5, 1], 1], [[2, 8, 3], -1], [[2, 9, 4], -1], [[3, 4, 10], 1], [[3, 6, 1], 1], [[3, 8, 2], 1], [[3, 10, 4], -1], [[4, 7, 1], 1], [[4, 9, 2], 1], [[4, 10, 3], 1], [[5, 6, 8], 1], [[5, 7, 9], 1], [[5, 8, 6], -1], [[5, 9, 7], -1], [[6, 7, 10], 1], [[6, 8, 5], 1], [[6, 10, 7], -1], [[7, 9, 5], 1], [[7, 10, 6], 1], [[8, 9, 10], 1], [[8, 10, 9], -1], [[9, 10, 8], 1]]])
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so5}}$ (2.2)

Define the representation space $V$.  We shall define the invariant tensors we need on $V.$

 so5 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5}\right],V\right)$
 ${\mathrm{frame name: V}}$ (2.3)

The standard inclusion of $\mathrm{so}\left(3\right)$in is given as the subalgebra of matrices which fix the vectors and  ${\mathrm{D}}_{\mathrm{x5}}$ .

 V > $\mathrm{Inv1}≔\left[\mathrm{D_x4},\mathrm{D_x5}\right]$
 ${\mathrm{Inv1}}{:=}\left[{\mathrm{D_x4}}{,}{\mathrm{D_x5}}\right]$ (2.4)
 V > $\mathrm{MatrixSubalgebra}\left(\mathrm{so5},A,\mathrm{Inv1}\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e5}}\right]$ (2.5)

Comparing with the matrices in A, we see this is precisely the subalgebra we want.

 V > $\left[{A}_{1},{A}_{2},{A}_{5}\right]$

We can define in as the subalgebra which preserves the subspaces spanned by] and . 



 so5 > $\mathrm{Inv2},\mathrm{Inv3}≔\left[\mathrm{D_x1},\mathrm{D_x2},\mathrm{D_x3}\right],\left[\mathrm{D_x4},\mathrm{D_x5}\right]$
 ${\mathrm{Inv2}}{,}{\mathrm{Inv3}}{:=}\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}\right]{,}\left[{\mathrm{D_x4}}{,}{\mathrm{D_x5}}\right]$ (2.6)
 > $\mathrm{MatrixSubalgebra}\left(\mathrm{so5},A,\left[\mathrm{Inv2},\mathrm{Inv3}\right]\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e5}}{,}{\mathrm{e10}}\right]$ (2.7)

Example 2.

The computation of Example 1 can be done with the other calling sequences.

1. With a representation.

 so5 > $\mathrm{ρ}≔\mathrm{Representation}\left(\mathrm{so5},V,A\right)$
 so5 > $\mathrm{MatrixSubalgebra}\left(\mathrm{ρ},\mathrm{Inv1}\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e5}}\right]$ (2.8)

2. With a Lie algebra of vector fields.

 V > $\mathrm{Gamma}≔\mathrm{evalDG}\left(\left[\mathrm{x2}\mathrm{D_x1}-\mathrm{x1}\mathrm{D_x2},\mathrm{x3}\mathrm{D_x1}-\mathrm{x1}\mathrm{D_x3},\mathrm{x4}\mathrm{D_x1}-\mathrm{x1}\mathrm{D_x4},\mathrm{x5}\mathrm{D_x1}-\mathrm{x1}\mathrm{D_x5},\mathrm{x3}\mathrm{D_x2}-\mathrm{x2}\mathrm{D_x3},\mathrm{x4}\mathrm{D_x2}-\mathrm{x2}\mathrm{D_x4},\mathrm{x5}\mathrm{D_x2}-\mathrm{x2}\mathrm{D_x5},\mathrm{x4}\mathrm{D_x3}-\mathrm{x3}\mathrm{D_x4},\mathrm{x5}\mathrm{D_x3}-\mathrm{x3}\mathrm{D_x5},\mathrm{x5}\mathrm{D_x4}-\mathrm{x4}\mathrm{D_x5}\right]\right)$
 ${\mathrm{Γ}}{:=}\left[{\mathrm{x2}}{}{\mathrm{D_x1}}{-}{\mathrm{x1}}{}{\mathrm{D_x2}}{,}{\mathrm{x3}}{}{\mathrm{D_x1}}{-}{\mathrm{x1}}{}{\mathrm{D_x3}}{,}{\mathrm{x4}}{}{\mathrm{D_x1}}{-}{\mathrm{x1}}{}{\mathrm{D_x4}}{,}{\mathrm{x5}}{}{\mathrm{D_x1}}{-}{\mathrm{x1}}{}{\mathrm{D_x5}}{,}{\mathrm{x3}}{}{\mathrm{D_x2}}{-}{\mathrm{x2}}{}{\mathrm{D_x3}}{,}{\mathrm{x4}}{}{\mathrm{D_x2}}{-}{\mathrm{x2}}{}{\mathrm{D_x4}}{,}{\mathrm{x5}}{}{\mathrm{D_x2}}{-}{\mathrm{x2}}{}{\mathrm{D_x5}}{,}{\mathrm{x4}}{}{\mathrm{D_x3}}{-}{\mathrm{x3}}{}{\mathrm{D_x4}}{,}{\mathrm{x5}}{}{\mathrm{D_x3}}{-}{\mathrm{x3}}{}{\mathrm{D_x5}}{,}{\mathrm{x5}}{}{\mathrm{D_x4}}{-}{\mathrm{x4}}{}{\mathrm{D_x5}}\right]$ (2.9)
 V > $\mathrm{MatrixSubalgebra}\left(\mathrm{so5},\mathrm{Gamma},\mathrm{Inv1}\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e5}}\right]$ (2.10)

3. With a Lie algebra constructed using the procedure SimpleLieAlgebraData .

 so5 > $\mathrm{LD1}≔\mathrm{SimpleLieAlgebraData}\left("so\left(5\right)",\mathrm{alg1}\right)$
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e8}}\right]$ (2.11)
 so5 > $\mathrm{DGsetup}\left(\mathrm{LD1}\right)$
 ${\mathrm{Lie algebra: alg1}}$ (2.12)
 alg > $\mathrm{MatrixSubalgebra}\left(\mathrm{alg1},\mathrm{Inv1}\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e5}}\right]$ (2.13)

Example 3.

Calculate the subalgebra of consisting of 2×2 block upper triangular matrices. First initialize the Lie algebra of all matrices. The labels 'E' and 'theta' must be unassigned names.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("gl\left(6\right)",\mathrm{gl6},\mathrm{labelformat}="gl",\mathrm{labels}=\left['E','\mathrm{θ}'\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right):$

Define the representation space.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5},\mathrm{x6}\right],\mathrm{V6}\right)$
 ${\mathrm{frame name: V6}}$ (2.14)

The matrices we want preserve the following subspaces of $V.$

 V6 > $\mathrm{Inv}≔\left[\left[\mathrm{D_x1},\mathrm{D_x2}\right],\left[\mathrm{D_x1},\mathrm{D_x2},\mathrm{D_x3},\mathrm{D_x4}\right]\right]$
 ${\mathrm{Inv}}{:=}\left[\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}\right]{,}\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]\right]$ (2.15)
 V6 > $A≔\mathrm{MatrixSubalgebra}\left(\mathrm{gl6},\mathrm{Inv}\right)$
 ${A}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}{,}{\mathrm{E15}}{,}{\mathrm{E16}}{,}{\mathrm{E21}}{,}{\mathrm{E22}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}{,}{\mathrm{E25}}{,}{\mathrm{E26}}{,}{\mathrm{E33}}{,}{\mathrm{E34}}{,}{\mathrm{E35}}{,}{\mathrm{E36}}{,}{\mathrm{E43}}{,}{\mathrm{E44}}{,}{\mathrm{E45}}{,}{\mathrm{E46}}{,}{\mathrm{E55}}{,}{\mathrm{E56}}{,}{\mathrm{E65}}{,}{\mathrm{E66}}\right]$ (2.16)

We can see what matrices these correspond to in several ways.  One method is to first form a general linear combination of the vectors in $A$.

 gl6 > $X≔\mathrm{DGzip}\left(\left[\mathrm{seq}\left(a||i,i=1..24\right)\right],A,"plus"\right)$
 ${X}{:=}{\mathrm{a1}}{}{\mathrm{E11}}{+}{\mathrm{a2}}{}{\mathrm{E12}}{+}{\mathrm{a3}}{}{\mathrm{E13}}{+}{\mathrm{a4}}{}{\mathrm{E14}}{+}{\mathrm{a5}}{}{\mathrm{E15}}{+}{\mathrm{a6}}{}{\mathrm{E16}}{+}{\mathrm{a7}}{}{\mathrm{E21}}{+}{\mathrm{a8}}{}{\mathrm{E22}}{+}{\mathrm{a9}}{}{\mathrm{E23}}{+}{\mathrm{a10}}{}{\mathrm{E24}}{+}{\mathrm{a11}}{}{\mathrm{E25}}{+}{\mathrm{a12}}{}{\mathrm{E26}}{+}{\mathrm{a13}}{}{\mathrm{E33}}{+}{\mathrm{a14}}{}{\mathrm{E34}}{+}{\mathrm{a15}}{}{\mathrm{E35}}{+}{\mathrm{a16}}{}{\mathrm{E36}}{+}{\mathrm{a17}}{}{\mathrm{E43}}{+}{\mathrm{a18}}{}{\mathrm{E44}}{+}{\mathrm{a19}}{}{\mathrm{E45}}{+}{\mathrm{a20}}{}{\mathrm{E46}}{+}{\mathrm{a21}}{}{\mathrm{E55}}{+}{\mathrm{a22}}{}{\mathrm{E56}}{+}{\mathrm{a23}}{}{\mathrm{E65}}{+}{\mathrm{a24}}{}{\mathrm{E66}}$ (2.17)

Now calculate the matrix associated to $X$  in the standard representation.

 gl6 > $\mathrm{StandardRepresentation}\left(\mathrm{gl6},X\right)$

Example 4.

In this example we calculate the intersection These are the skew-symmetric matrices which also preserve a non-degenerate 2-form. Then we show that this intersection is isomorphic to  First we initialize the Lie algebra for  The labels 'R' and 'sigma' must be unassigned names.

 gl6 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(8\right)",\mathrm{so8},\mathrm{labelformat}="gl",\mathrm{labels}=\left['R','\mathrm{σ}'\right]\right):$
 gl6 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so8}}$ (2.18)

Now define an 8-dimensional representation space and a 2-form on $V.$

 so8 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5},\mathrm{x6},\mathrm{x7},\mathrm{x8}\right],\mathrm{V8}\right)$
 ${\mathrm{frame name: V8}}$ (2.19)
 V8 > $\mathrm{Ω}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx5}+\mathrm{dx2}&w\mathrm{dx6}+\mathrm{dx3}&w\mathrm{dx7}+\mathrm{dx4}&w\mathrm{dx8}\right)$
 ${\mathrm{Ω}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx5}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx6}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx8}}$ (2.20)

Find the subalgebra of $\mathrm{so}\left(8\right)$ which preserves this 2-form.

 V8 > $H≔\mathrm{MatrixSubalgebra}\left(\mathrm{so8},\left[\mathrm{Ω}\right]\right)$
 ${H}{:=}\left[{\mathrm{R12}}{+}{\mathrm{R56}}{,}{\mathrm{R13}}{+}{\mathrm{R57}}{,}{\mathrm{R14}}{+}{\mathrm{R58}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{+}{\mathrm{R25}}{,}{\mathrm{R17}}{+}{\mathrm{R35}}{,}{\mathrm{R18}}{+}{\mathrm{R45}}{,}{\mathrm{R23}}{+}{\mathrm{R67}}{,}{\mathrm{R24}}{+}{\mathrm{R68}}{,}{\mathrm{R26}}{,}{\mathrm{R27}}{+}{\mathrm{R36}}{,}{\mathrm{R28}}{+}{\mathrm{R46}}{,}{\mathrm{R34}}{+}{\mathrm{R78}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{+}{\mathrm{R47}}{,}{\mathrm{R48}}\right]$ (2.21)

Here are the explicit matrices.

 so8 > $M≔\mathrm{map2}\left(\mathrm{StandardRepresentation},\mathrm{so8},H\right)$

Check that the matrices belong to $\mathrm{so}\left(8\right).$



 so8 > $\mathrm{Query}\left(M,"so\left(8\right)","MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.22)

Check that the matrices belong to $\mathrm{sp}\left(8,\mathrm{ℝ}\right)$.

 so8 > $\mathrm{Query}\left(M,"sp\left(8, R\right)","MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.23)

The isomorphism to $u\left(4\right)$ is given by  ${\mathrm{Φ}}_{}\left(\left[\begin{array}{rr}A& B\\ C& \mathrm{D}\end{array}\right]\right)$ =  where  are 4×4 matrices.  We use the command SubMatrix  to construct this map.

 so8 > $\mathrm{Φ}≔X→\mathrm{LinearAlgebra}:-\mathrm{SubMatrix}\left(X,1..4,1..4\right)+I\mathrm{LinearAlgebra}:-\mathrm{SubMatrix}\left(X,5..8,1..4\right)$
 ${\mathrm{Φ}}{:=}{X}{→}{\mathrm{LinearAlgebra}}{:-}{\mathrm{SubMatrix}}{}\left({X}{,}{1}{..}{4}{,}{1}{..}{4}\right){+}{I}{}{\mathrm{LinearAlgebra}}{:-}{\mathrm{SubMatrix}}{}\left({X}{,}{5}{..}{8}{,}{1}{..}{4}\right)$ (2.24)
 so8 > $N≔\mathrm{map}\left(\mathrm{Φ},M\right)$

Check that each of these matrices belong to $u\left(4\right)$.

 so8 > $\mathrm{Query}\left(N,"u\left(4\right)","MatrixAlgebra"\right)$
 ${\mathrm{true}}$ (2.25)

Finally, we see that the structure equations for these two matrix algebras are identical.

 so8 > $\mathrm{LieAlgebraData}\left(M,\mathrm{alg1}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e16}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e10}}{+}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e10}}{+}{2}{}{\mathrm{e16}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e15}}\right]{=}{-}{2}{}{\mathrm{e14}}{+}{2}{}{\mathrm{e16}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e13}}\right]$ (2.26)
 so8 > $\mathrm{LieAlgebraData}\left(N,\mathrm{alg2},\mathrm{method}="real"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e16}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e10}}{+}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e10}}{+}{2}{}{\mathrm{e16}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e15}}\right]{=}{-}{2}{}{\mathrm{e14}}{+}{2}{}{\mathrm{e16}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e13}}\right]$ (2.27)

Example 5.

The compact real form of the exceptional Lie algebra as the subalgebra of can be computed using the command MatrixAlgebras. First we initialize the Lie algebra  $\mathrm{so}\left(7\right).$



 so7 > $\mathrm{RemoveFrame}\left(\mathrm{so8}\right):$
 gl6 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(7\right)",\mathrm{so7},\mathrm{labelformat}="gl",\mathrm{labels}=\left['R','\mathrm{σ}'\right]\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e19}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e20}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e19}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e20}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e19}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e20}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e19}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e20}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e17}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e18}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e18}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e16}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e19}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e21}}{,}\left[{\mathrm{e19}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e20}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e19}}\right]{,}\left[{\mathrm{R12}}{,}{\mathrm{R13}}{,}{\mathrm{R14}}{,}{\mathrm{R15}}{,}{\mathrm{R16}}{,}{\mathrm{R17}}{,}{\mathrm{R23}}{,}{\mathrm{R24}}{,}{\mathrm{R25}}{,}{\mathrm{R26}}{,}{\mathrm{R27}}{,}{\mathrm{R34}}{,}{\mathrm{R35}}{,}{\mathrm{R36}}{,}{\mathrm{R37}}{,}{\mathrm{R45}}{,}{\mathrm{R46}}{,}{\mathrm{R47}}{,}{\mathrm{R56}}{,}{\mathrm{R57}}{,}{\mathrm{R67}}\right]{,}\left[{\mathrm{σ12}}{,}{\mathrm{σ13}}{,}{\mathrm{σ14}}{,}{\mathrm{σ15}}{,}{\mathrm{σ16}}{,}{\mathrm{σ17}}{,}{\mathrm{σ23}}{,}{\mathrm{σ24}}{,}{\mathrm{σ25}}{,}{\mathrm{σ26}}{,}{\mathrm{σ27}}{,}{\mathrm{σ34}}{,}{\mathrm{σ35}}{,}{\mathrm{σ36}}{,}{\mathrm{σ37}}{,}{\mathrm{σ45}}{,}{\mathrm{σ46}}{,}{\mathrm{σ47}}{,}{\mathrm{σ56}}{,}{\mathrm{σ57}}{,}{\mathrm{σ67}}\right]$ (2.28)
 gl6 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so7}}$ (2.29)

Now define a7-dimensional representation space and a 3-form on $V.$

 so8 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5},\mathrm{x6},\mathrm{x7}\right],\mathrm{V7}\right)$
 ${\mathrm{frame name: V7}}$ (2.30)
 V7 > $\mathrm{σ1}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx3}-\mathrm{dx2}&w\mathrm{dx4}\right)$
 ${\mathrm{σ1}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}$ (2.31)
 V7 > $\mathrm{σ2}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx4}+\mathrm{dx2}&w\mathrm{dx3}\right)$
 ${\mathrm{σ2}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}$ (2.32)
 V7 > $\mathrm{σ3}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx2}+\mathrm{dx3}&w\mathrm{dx4}\right)$
 ${\mathrm{σ3}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}$ (2.33)
 so8 > $\mathrm{Φ}≔\mathrm{evalDG}\left(\mathrm{σ1}&w\mathrm{dx5}-\mathrm{σ2}&w\mathrm{dx6}+\mathrm{σ3}&w\mathrm{dx7}+\left(\mathrm{dx5}&w\mathrm{dx6}\right)&w\mathrm{dx7}\right)$
 ${\mathrm{Φ}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx5}}{-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx6}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx6}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx5}}{}{\bigwedge }{}{\mathrm{dx6}}{}{\bigwedge }{}{\mathrm{dx7}}$ (2.34)

Calculate the subalgebra of  $\mathrm{so}\left(7\right)$ which leaves the 3-form invariant.

 V8 > $\mathrm{G2}≔\mathrm{MatrixSubalgebra}\left(\mathrm{so7},\left[\mathrm{Φ}\right]\right)$
 ${\mathrm{G2}}{:=}\left[{\mathrm{R12}}{-}{\mathrm{R56}}{,}{\mathrm{R13}}{-}{\mathrm{R67}}{,}{\mathrm{R14}}{-}{\mathrm{R57}}{,}{\mathrm{R15}}{+}{\mathrm{R47}}{,}{\mathrm{R16}}{+}{\mathrm{R37}}{,}{\mathrm{R17}}{-}{\mathrm{R45}}{,}{\mathrm{R23}}{-}{\mathrm{R57}}{,}{\mathrm{R24}}{+}{\mathrm{R67}}{,}{\mathrm{R25}}{+}{\mathrm{R37}}{,}{\mathrm{R26}}{-}{\mathrm{R47}}{,}{\mathrm{R27}}{+}{\mathrm{R46}}{,}{\mathrm{R34}}{-}{\mathrm{R56}}{,}{\mathrm{R35}}{+}{\mathrm{R46}}{,}{\mathrm{R36}}{-}{\mathrm{R45}}\right]$ (2.35)

Here are the explicit matrices.

 so8 > $\mathrm{M2}≔\mathrm{map2}\left(\mathrm{StandardRepresentation},\mathrm{so7},\mathrm{G2}\right)$

The Lie algebra defined by either the vectors or the matrices $\mathrm{M2}$  is a 14-dimensional Lie algebra with negative-definite Killing form and 2-dimensional Cartan subalgebra.

 > $\mathrm{LD2}≔\mathrm{LieAlgebraData}\left(\mathrm{G2},\mathrm{g2}\right):$
 so7 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: g2}}$ (2.36)
 so7 > $\mathrm{LinearAlgebra}:-\mathrm{IsDefinite}\left(-\mathrm{Killing}\left(\right)\right)$
 ${\mathrm{true}}$ (2.37)
 g2 > $\mathrm{CartanSubalgebra}\left(\right)$
 $\left[{\mathrm{e1}}{,}{\mathrm{e4}}{-}{\mathrm{e10}}\right]$ (2.38)

Example 6.

The split real form of the exceptional Lie algebra as the subalgebra of $\mathrm{so}$(4, 3)  is similarly computed.

 so7 > $\mathrm{RemoveFrame}\left(\mathrm{so7}\right)$
 ${8}$ (2.39)
 gl6 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(4, 3\right)",\mathrm{so43},\mathrm{version}=2,\mathrm{labelformat}="gl",\mathrm{labels}=\left['R','\mathrm{σ}'\right]\right):$
 gl6 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so43}}$ (2.40)

Now define a 7-dimensional representation space and a 3-form on $V.$

 so8 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4},\mathrm{x5},\mathrm{x6},\mathrm{x7}\right],\mathrm{V7}\right)$
 ${\mathrm{frame name: V7}}$ (2.41)
 V7 > $\mathrm{σ1}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx3}-\mathrm{dx2}&w\mathrm{dx4}\right)$
 ${\mathrm{σ1}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}$ (2.42)
 V7 > $\mathrm{σ2}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx4}+\mathrm{dx2}&w\mathrm{dx3}\right)$
 ${\mathrm{σ2}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}$ (2.43)
 V7 > $\mathrm{σ3}≔\mathrm{evalDG}\left(\mathrm{dx1}&w\mathrm{dx2}+\mathrm{dx3}&w\mathrm{dx4}\right)$
 ${\mathrm{σ3}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}$ (2.44)
 so8 > $\mathrm{Φ}≔\mathrm{evalDG}\left(\mathrm{σ1}&w\mathrm{dx5}-\mathrm{σ2}&w\mathrm{dx6}+\mathrm{σ3}&w\mathrm{dx7}-\left(\mathrm{dx5}&w\mathrm{dx6}\right)&w\mathrm{dx7}\right)$
 ${\mathrm{Φ}}{:=}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx5}}{-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx6}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx6}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx7}}{-}{\mathrm{dx5}}{}{\bigwedge }{}{\mathrm{dx6}}{}{\bigwedge }{}{\mathrm{dx7}}$ (2.45)

Calculate the subalgebra of $\mathrm{so}\left(7\right)$ which  leaves the 3-form invariant.

 V8 > $\mathrm{G2}≔\mathrm{MatrixSubalgebra}\left(\mathrm{so43},\left[\mathrm{Φ}\right]\right)$
 ${\mathrm{G2}}{:=}\left[{\mathrm{R12}}{-}{\mathrm{R56}}{,}{\mathrm{R13}}{-}{\mathrm{R67}}{,}{\mathrm{R14}}{-}{\mathrm{R57}}{,}{\mathrm{R23}}{-}{\mathrm{R57}}{,}{\mathrm{R24}}{+}{\mathrm{R67}}{,}{\mathrm{R34}}{-}{\mathrm{R56}}{,}{\mathrm{R15}}{+}{\mathrm{R47}}{,}{\mathrm{R16}}{+}{\mathrm{R37}}{,}{\mathrm{R17}}{-}{\mathrm{R45}}{,}{\mathrm{R25}}{+}{\mathrm{R37}}{,}{\mathrm{R26}}{-}{\mathrm{R47}}{,}{\mathrm{R27}}{+}{\mathrm{R46}}{,}{\mathrm{R35}}{+}{\mathrm{R46}}{,}{\mathrm{R36}}{-}{\mathrm{R45}}\right]$ (2.46)

Here are the explicit matrices.

 so8 > $\mathrm{M2}≔\mathrm{map2}\left(\mathrm{StandardRepresentation},\mathrm{so43},\mathrm{G2}\right)$

The Lie algebra defined by either the vectors or the matrices $\mathrm{M2}$  is a 14-dimensional Lie algebra.

 V7 > $\mathrm{LD6}≔\mathrm{LieAlgebraData}\left(\mathrm{G2},\mathrm{g2S}\right)$
 ${\mathrm{LD6}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{+}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{+}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{-}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e8}}{+}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e9}}{+}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{+}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e12}}{+}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e8}}{+}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e12}}{+}{\mathrm{e13}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e12}}{+}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{-}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{+}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e11}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e3}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e2}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{2}{}{\mathrm{e6}}\right]$ (2.47)
 so7 > $\mathrm{DGsetup}\left(\mathrm{LD6}\right)$
 ${\mathrm{Lie algebra: g2S}}$ (2.48)
 g2S > $B≔\mathrm{KillingForm}\left(\right)$
 ${B}{:=}{-}{8}{}{\mathrm{θ1}}{}{\mathrm{θ6}}{-}{8}{}{\mathrm{θ7}}{}{\mathrm{θ11}}{+}{16}{}{\mathrm{θ8}}{}{\mathrm{θ8}}{+}{8}{}{\mathrm{θ5}}{}{\mathrm{θ2}}{-}{16}{}{\mathrm{θ5}}{}{\mathrm{θ5}}{+}{8}{}{\mathrm{θ8}}{}{\mathrm{θ10}}{+}{16}{}{\mathrm{θ9}}{}{\mathrm{θ9}}{+}{16}{}{\mathrm{θ12}}{}{\mathrm{θ12}}{+}{8}{}{\mathrm{θ12}}{}{\mathrm{θ13}}{-}{16}{}{\mathrm{θ1}}{}{\mathrm{θ1}}{-}{8}{}{\mathrm{θ4}}{}{\mathrm{θ3}}{-}{16}{}{\mathrm{θ4}}{}{\mathrm{θ4}}{+}{8}{}{\mathrm{θ9}}{}{\mathrm{θ14}}{+}{8}{}{\mathrm{θ10}}{}{\mathrm{θ8}}{+}{16}{}{\mathrm{θ10}}{}{\mathrm{θ10}}{-}{8}{}{\mathrm{θ11}}{}{\mathrm{θ7}}{+}{16}{}{\mathrm{θ11}}{}{\mathrm{θ11}}{+}{8}{}{\mathrm{θ13}}{}{\mathrm{θ12}}{+}{16}{}{\mathrm{θ13}}{}{\mathrm{θ13}}{-}{16}{}{\mathrm{θ3}}{}{\mathrm{θ3}}{-}{8}{}{\mathrm{θ3}}{}{\mathrm{θ4}}{+}{8}{}{\mathrm{θ14}}{}{\mathrm{θ9}}{+}{16}{}{\mathrm{θ14}}{}{\mathrm{θ14}}{-}{16}{}{\mathrm{θ2}}{}{\mathrm{θ2}}{+}{8}{}{\mathrm{θ2}}{}{\mathrm{θ5}}{-}{8}{}{\mathrm{θ6}}{}{\mathrm{θ1}}{-}{16}{}{\mathrm{θ6}}{}{\mathrm{θ6}}{+}{16}{}{\mathrm{θ7}}{}{\mathrm{θ7}}$ (2.49)
 g2S > $\mathrm{Tensor}:-\mathrm{QuadraticFormSignature}\left(B\right)$
 $\left[\left[{\mathrm{e7}}{,}{\mathrm{e7}}{+}{2}{}{\mathrm{e11}}{,}{\mathrm{e8}}{,}{\mathrm{e8}}{-}{2}{}{\mathrm{e10}}{,}{\mathrm{e9}}{,}{\mathrm{e9}}{-}{2}{}{\mathrm{e14}}{,}{\mathrm{e12}}{,}{\mathrm{e12}}{-}{2}{}{\mathrm{e13}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e1}}{-}{2}{}{\mathrm{e6}}{,}{\mathrm{e2}}{,}{\mathrm{e2}}{+}{2}{}{\mathrm{e5}}{,}{\mathrm{e3}}{,}{\mathrm{e3}}{-}{2}{}{\mathrm{e4}}\right]{,}\left[{}\right]\right]$ (2.50)
 g2S > $\mathrm{map}\left(\mathrm{nops},%\right)$
 $\left[{8}{,}{6}{,}{0}\right]$ (2.51)