GradeSemiSimpleLieAlgebra - Maple Help

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LieAlgebras[GradeSemiSimpleLieAlgebra] - find the grading of a semi-simple Lie algebra defined by a set of simple roots or restricted simple roots

Calling Sequences

GradeSemiSimpleLieAlgebra(T2, method = "non-compact")

Parameters

$\mathrm{Σ}$       - a list or set of column vectors, defining a subset of the simple roots or a subset of the restricted simple roots

T1      - a table, with indices that include "RootSpaceDecomposition", "CartanSubalgebra", "SimpleRoots", "PositiveRoots"

T2      - a table, with indices that include "RestrictedRootSpaceDecomposition", "CartanSubalgebra", "RestrictedSimpleRoots", "RestrictedPositiveRoots"

Description

 • Let g be a Lie algebra. A grading of g is a (vector space) direct sum decomposition g =where Gradings of semi-simple Lie algebras can easily be constructed from the root space decomposition. Let h be a Cartan subalgebra and  the associated root space decomposition Let be a choice of positive roots and let be a set of simple roots. Every root α is a sum of simple roots, say and one defines the height of the root as ht.
 • Now let be a collection of simple roots and define the Σ height of as ht ${}_{\mathrm{Σ}}$where the sum is taken over those such that . Then the subspaces

  and  

define a (symmetric) grading g =

 • For real Lie algebras, real gradings can be similarly constructed using the restricted root space decomposition.
 • The command Query/"Gradation" will test if a given decomposition of a Lie algebra is graded.

Examples

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

Example 1.

We calculate the various gradations for We use the command SimpleLieAlgebraData to initialize the Lie algebra.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("sl\left(4\right)",\mathrm{sl4},\mathrm{labelformat}="gl",\mathrm{labels}=\left[E,\mathrm{\omega }\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sl4}}$ (2.1)
 sl4 > $P≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$

We use the command SimpleLieAlgebraProperties to create a table containing the structure properties of $\mathrm{sl}\left(4\right)$.

 > $T≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{sl4}\right):$
 sl4 > $\mathrm{SR}≔T\left["SimpleRoots"\right]$
 ${\mathrm{SR}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]$ (2.2)

Here are the possible subsets of the set of simple roots.

 sl4 > $\mathrm{\Sigma }≔\left[\left[\right],\mathrm{SR}\left[1..1\right],\mathrm{SR}\left[2..2\right],\mathrm{SR}\left[3..3\right],\mathrm{SR}\left[1..2\right],\mathrm{SR}\left[2..3\right],\left[\mathrm{SR}\left[1\right],\mathrm{SR}\left[3\right]\right],\mathrm{SR}\right]$
 ${\mathrm{Σ}}{:=}\left[\left[{}\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]\right]$ (2.3)

Here are the gradings defined by each subset of the simple roots.

 sl4 > $\mathrm{\Sigma }\left[1\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[1\right],P\right)$
 $\left[{}\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}{,}{\mathrm{E14}}{,}{\mathrm{E21}}{,}{\mathrm{E32}}{,}{\mathrm{E43}}{,}{\mathrm{E31}}{,}{\mathrm{E42}}{,}{\mathrm{E41}}\right]\right]\right)$ (2.4)
 sl4 > $\mathrm{\Sigma }\left[2\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[2\right],P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E23}}{,}{\mathrm{E34}}{,}{\mathrm{E24}}{,}{\mathrm{E32}}{,}{\mathrm{E43}}{,}{\mathrm{E42}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E31}}{,}{\mathrm{E41}}\right]\right]\right)$ (2.5)
 sl4 > $\mathrm{\Sigma }\left[3\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[3\right],P\right)$
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E34}}{,}{\mathrm{E21}}{,}{\mathrm{E43}}\right]{,}{1}{=}\left[{\mathrm{E23}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}{,}{\mathrm{E14}}\right]{,}{-}{1}{=}\left[{\mathrm{E32}}{,}{\mathrm{E31}}{,}{\mathrm{E42}}{,}{\mathrm{E41}}\right]\right]\right)$ (2.6)
 sl4 > $\mathrm{\Sigma }\left[4\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[4\right],P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E13}}{,}{\mathrm{E21}}{,}{\mathrm{E32}}{,}{\mathrm{E31}}\right]{,}{1}{=}\left[{\mathrm{E34}}{,}{\mathrm{E24}}{,}{\mathrm{E14}}\right]{,}{-}{1}{=}\left[{\mathrm{E43}}{,}{\mathrm{E42}}{,}{\mathrm{E41}}\right]\right]\right)$ (2.7)
 sl4 > $\mathrm{\Sigma }\left[5\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[5\right],P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E34}}{,}{\mathrm{E43}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E24}}\right]{,}{2}{=}\left[{\mathrm{E13}}{,}{\mathrm{E14}}\right]{,}{-}{2}{=}\left[{\mathrm{E31}}{,}{\mathrm{E41}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E32}}{,}{\mathrm{E42}}\right]\right]\right)$ (2.8)
 sl4 > $\mathrm{\Sigma }\left[6\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[6\right],P\right)$
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E12}}{,}{\mathrm{E21}}\right]{,}{1}{=}\left[{\mathrm{E23}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}\right]{,}{2}{=}\left[{\mathrm{E24}}{,}{\mathrm{E14}}\right]{,}{-}{2}{=}\left[{\mathrm{E42}}{,}{\mathrm{E41}}\right]{,}{-}{1}{=}\left[{\mathrm{E32}}{,}{\mathrm{E43}}{,}{\mathrm{E31}}\right]\right]\right)$ (2.9)
 sl4 > $\mathrm{\Sigma }\left[7\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[7\right],P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E23}}{,}{\mathrm{E32}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}\right]{,}{2}{=}\left[{\mathrm{E14}}\right]{,}{-}{2}{=}\left[{\mathrm{E41}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E43}}{,}{\mathrm{E31}}{,}{\mathrm{E42}}\right]\right]\right)$ (2.10)
 sl4 > $\mathrm{\Sigma }\left[8\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[8\right],P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {1}\\ {2}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E34}}\right]{,}{2}{=}\left[{\mathrm{E13}}{,}{\mathrm{E24}}\right]{,}{3}{=}\left[{\mathrm{E14}}\right]{,}{-}{3}{=}\left[{\mathrm{E41}}\right]{,}{-}{2}{=}\left[{\mathrm{E31}}{,}{\mathrm{E42}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E32}}{,}{\mathrm{E43}}\right]\right]\right)$ (2.11)
 sl4 > $\mathrm{\Sigma }\left[2\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[2\right],P\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E23}}{,}{\mathrm{E34}}{,}{\mathrm{E24}}{,}{\mathrm{E32}}{,}{\mathrm{E43}}{,}{\mathrm{E42}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E13}}{,}{\mathrm{E14}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E31}}{,}{\mathrm{E41}}\right]\right]\right)$ (2.12)

The Query command can be used to check that each of these define a grading of $\mathrm{sl}\left(4\right)$.

 sl4 > $\mathrm{G7}≔\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[7\right],P\right)$
 ${\mathrm{G7}}{:=}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E33}}{,}{\mathrm{E23}}{,}{\mathrm{E32}}\right]{,}{1}{=}\left[{\mathrm{E12}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}\right]{,}{2}{=}\left[{\mathrm{E14}}\right]{,}{-}{2}{=}\left[{\mathrm{E41}}\right]{,}{-}{1}{=}\left[{\mathrm{E21}}{,}{\mathrm{E43}}{,}{\mathrm{E31}}{,}{\mathrm{E42}}\right]\right]\right)$ (2.13)
 sl4 > $\mathrm{Query}\left(\mathrm{G7},"Gradation"\right)$
 ${\mathrm{true}}$ (2.14)

Example 2.

We calculate the various gradings for We use the command SimpleLieAlgebraData to initialize the Lie algebra.

 sl4 > $\mathrm{LD2}≔\mathrm{SimpleLieAlgebraData}\left("so\left(5,3\right)",\mathrm{so53},\mathrm{labelformat}="gl",\mathrm{labels}=\left[R,\mathrm{\theta }\right]\right):$
 sl4 > $\mathrm{DGsetup}\left(\mathrm{LD2}\right)$
 ${\mathrm{Lie algebra: so53}}$ (2.15)



We use the command SimpleLieAlgebraProperties to calculate the restricted root space decomposition, restricted simple roots, etc.

 so53 > $T≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{so53}\right):$
 so53 > $\mathrm{RSR}≔T\left["RestrictedSimpleRoots"\right]$
 ${\mathrm{RSR}}{:=}\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]$ (2.16)

The subsets of the restricted simple roots are:

 so53 > $\mathrm{\Sigma }≔\left[\mathrm{RSR},\mathrm{RSR}\left[1..2\right],\mathrm{RSR}\left[2..3\right],\left[\mathrm{RSR}\left[1\right],\mathrm{RSR}\left[3\right]\right],\mathrm{RSR}\left[1..1\right],\mathrm{RSR}\left[2..2\right],\mathrm{RSR}\left[3..3\right],\left[\right]\right]$

Here are the possible gradings for

 so53 > $\mathrm{\Sigma }\left[1\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[1\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R78}}{,}{\mathrm{R33}}\right]{,}{1}{=}\left[{\mathrm{R12}}{,}{\mathrm{R23}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}{2}{=}\left[{\mathrm{R13}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}\right]{,}{3}{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}\right]{,}{5}{=}\left[{\mathrm{R15}}\right]{,}{4}{=}\left[{\mathrm{R16}}\right]{,}{-}{5}{=}\left[{\mathrm{R42}}\right]{,}{-}{4}{=}\left[{\mathrm{R43}}\right]{,}{-}{3}{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}\right]{,}{-}{2}{=}\left[{\mathrm{R31}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}\right]{,}{-}{1}{=}\left[{\mathrm{R21}}{,}{\mathrm{R32}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.17)
 so53 > $\mathrm{\Sigma }\left[2\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[2\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]{,}{1}{=}\left[{\mathrm{R26}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R12}}{,}{\mathrm{R23}}\right]{,}{2}{=}\left[{\mathrm{R16}}{,}{\mathrm{R13}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}\right]{,}{3}{=}\left[{\mathrm{R15}}\right]{,}{-}{3}{=}\left[{\mathrm{R42}}\right]{,}{-}{2}{=}\left[{\mathrm{R43}}{,}{\mathrm{R31}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}\right]{,}{-}{1}{=}\left[{\mathrm{R53}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R21}}{,}{\mathrm{R32}}\right]\right]\right)$ (2.18)
 so53 > $\mathrm{\Sigma }\left[3\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[3\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R21}}\right]{,}{1}{=}\left[{\mathrm{R13}}{,}{\mathrm{R23}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}{2}{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}\right]{,}{3}{=}\left[{\mathrm{R16}}{,}{\mathrm{R26}}\right]{,}{4}{=}\left[{\mathrm{R15}}\right]{,}{-}{4}{=}\left[{\mathrm{R42}}\right]{,}{-}{3}{=}\left[{\mathrm{R43}}{,}{\mathrm{R53}}\right]{,}{-}{2}{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}\right]{,}{-}{1}{=}\left[{\mathrm{R31}}{,}{\mathrm{R32}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.19)
 so53 > $\mathrm{\Sigma }\left[4\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[4\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R23}}{,}{\mathrm{R32}}\right]{,}{1}{=}\left[{\mathrm{R13}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R12}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}{2}{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}\right]{,}{3}{=}\left[{\mathrm{R16}}{,}{\mathrm{R15}}\right]{,}{-}{3}{=}\left[{\mathrm{R43}}{,}{\mathrm{R42}}\right]{,}{-}{2}{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}\right]{,}{-}{1}{=}\left[{\mathrm{R31}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R21}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.20)
 so53 > $\mathrm{\Sigma }\left[5\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[5\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{1}\\ {-}{1}\\ {0}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R26}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R23}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R53}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R32}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]{,}{1}{=}\left[{\mathrm{R16}}{,}{\mathrm{R13}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R12}}{,}{\mathrm{R15}}\right]{,}{-}{1}{=}\left[{\mathrm{R43}}{,}{\mathrm{R31}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R21}}{,}{\mathrm{R42}}\right]\right]\right)$ (2.21)
 so53 > $\mathrm{\Sigma }\left[6\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[6\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{0}\\ {1}\\ {-}{1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R12}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R21}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]{,}{1}{=}\left[{\mathrm{R16}}{,}{\mathrm{R13}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R23}}\right]{,}{2}{=}\left[{\mathrm{R15}}\right]{,}{-}{2}{=}\left[{\mathrm{R42}}\right]{,}{-}{1}{=}\left[{\mathrm{R43}}{,}{\mathrm{R31}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R32}}\right]\right]\right)$ (2.22)
 so53 > $\mathrm{\Sigma }\left[7\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[7\right],T,\mathrm{method}="non-compact"\right)$
 $\left[\left[\begin{array}{r}{0}\\ {0}\\ {1}\end{array}\right]\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R13}}{,}{\mathrm{R12}}{,}{\mathrm{R23}}{,}{\mathrm{R31}}{,}{\mathrm{R21}}{,}{\mathrm{R32}}\right]{,}{1}{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}{2}{=}\left[{\mathrm{R16}}{,}{\mathrm{R26}}{,}{\mathrm{R15}}\right]{,}{-}{2}{=}\left[{\mathrm{R43}}{,}{\mathrm{R53}}{,}{\mathrm{R42}}\right]{,}{-}{1}{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.23)
 so53 > $\mathrm{\Sigma }\left[8\right],\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[8\right],T,\mathrm{method}="non-compact"\right)$
 $\left[{}\right]{,}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}{,}{\mathrm{R16}}{,}{\mathrm{R13}}{,}{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}{,}{\mathrm{R12}}{,}{\mathrm{R23}}{,}{\mathrm{R15}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}{,}{\mathrm{R43}}{,}{\mathrm{R31}}{,}{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}{,}{\mathrm{R21}}{,}{\mathrm{R32}}{,}{\mathrm{R42}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.24)

The Query command can be used to check that each of these define a grading of $\mathrm{so}\left(5,3\right)$.

 so53 > $\mathrm{G1}≔\mathrm{GradeSemiSimpleLieAlgebra}\left(\mathrm{\Sigma }\left[1\right],T,\mathrm{method}="non-compact"\right)$
 ${\mathrm{G1}}{:=}{\mathrm{table}}\left(\left[{0}{=}\left[{\mathrm{R78}}{,}{\mathrm{R33}}{,}{\mathrm{R22}}{,}{\mathrm{R11}}\right]{,}{1}{=}\left[{\mathrm{R12}}{,}{\mathrm{R23}}{,}{\mathrm{R37}}{,}{\mathrm{R38}}\right]{,}{2}{=}\left[{\mathrm{R13}}{,}{\mathrm{R27}}{,}{\mathrm{R28}}\right]{,}{3}{=}\left[{\mathrm{R17}}{,}{\mathrm{R18}}{,}{\mathrm{R26}}\right]{,}{5}{=}\left[{\mathrm{R15}}\right]{,}{4}{=}\left[{\mathrm{R16}}\right]{,}{-}{5}{=}\left[{\mathrm{R42}}\right]{,}{-}{4}{=}\left[{\mathrm{R43}}\right]{,}{-}{3}{=}\left[{\mathrm{R47}}{,}{\mathrm{R48}}{,}{\mathrm{R53}}\right]{,}{-}{2}{=}\left[{\mathrm{R31}}{,}{\mathrm{R57}}{,}{\mathrm{R58}}\right]{,}{-}{1}{=}\left[{\mathrm{R21}}{,}{\mathrm{R32}}{,}{\mathrm{R67}}{,}{\mathrm{R68}}\right]\right]\right)$ (2.25)
 so53 > $\mathrm{Query}\left(\mathrm{G1},"Gradation"\right)$
 ${\mathrm{true}}$ (2.26)