transform a Cartan matrix to standard form - Maple Help

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LieAlgebras[CartanMatrixToStandardForm] - transform a Cartan matrix to standard form

Calling Sequences

CartanMatrixToStandardForm(C,${}$SR)

Parameters

C   - a square matrix

SR  - (optional) a list of vectors, the simple roots used to determine the Cartan matrix for a simple Lie algebra

Description

 • Let be a set of simple roots for g. Then the associated Cartan matrix is the $m×m$ matrix with entries

.

(See CartanMatrix for the definition of the vectors ${H}_{{\mathrm{α}}_{i}}$ )

 • A permutation of the roots leads to a different but equivalent Cartan matrix.
 • The command CartanMatrixToStandardForm transforms a Cartan matrix to the standard form for each root type.
 • The command returns the Cartan matrix in standard form, a permutation matrix, and a string denoting the root type. The permutation matrix will transform the given Cartan matrix to its standard form by a similarity transformation.
 • If the second calling is invoked, then the second element of the output is the permuted set of simple roots which will generate the standard form of the Cartan matrix.

Examples

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

Example 1.

We define 4 different Cartan matrices and calculate their standard forms and root type.

 > $\mathrm{CM1}≔\mathrm{Matrix}\left(\left[\left[2,-1,0,-1,-1,0\right],\left[-1,2,0,0,0,0\right],\left[0,0,2,0,0,-1\right],\left[-1,0,0,2,0,-1\right],\left[-1,0,0,0,2,0\right],\left[0,0,-1,-1,0,2\right]\right]\right)$
 ${\mathrm{CM1}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {-}{1}& {-}{1}& {0}\\ {-}{1}& {2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {2}& {0}& {0}& {-}{1}\\ {-}{1}& {0}& {0}& {2}& {0}& {-}{1}\\ {-}{1}& {0}& {0}& {0}& {2}& {0}\\ {0}& {0}& {-}{1}& {-}{1}& {0}& {2}\end{array}\right]$ (2.1)
 > $\mathrm{CM2}≔\mathrm{Matrix}\left(\left[\left[2,0,0,0,-1,0\right],\left[0,2,-1,0,0,0\right],\left[0,-1,2,0,-1,-1\right],\left[0,0,0,2,0,-1\right],\left[-1,0,-1,0,2,0\right],\left[0,0,-1,-1,0,2\right]\right]\right)$
 ${\mathrm{CM2}}{:=}\left[\begin{array}{rrrrrr}{2}& {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {0}& {-}{1}& {-}{1}\\ {0}& {0}& {0}& {2}& {0}& {-}{1}\\ {-}{1}& {0}& {-}{1}& {0}& {2}& {0}\\ {0}& {0}& {-}{1}& {-}{1}& {0}& {2}\end{array}\right]$ (2.2)
 > $\mathrm{CM3}≔\mathrm{Matrix}\left(\left[\left[2,0,0,-1,-1,0\right],\left[0,2,0,-1,0,-1\right],\left[0,0,2,0,-2,0\right],\left[-1,-1,0,2,0,0\right],\left[-1,0,-1,0,2,0\right],\left[0,-1,0,0,0,2\right]\right]\right)$
 ${\mathrm{CM3}}{:=}\left[\begin{array}{rrrrrr}{2}& {0}& {0}& {-}{1}& {-}{1}& {0}\\ {0}& {2}& {0}& {-}{1}& {0}& {-}{1}\\ {0}& {0}& {2}& {0}& {-}{2}& {0}\\ {-}{1}& {-}{1}& {0}& {2}& {0}& {0}\\ {-}{1}& {0}& {-}{1}& {0}& {2}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {2}\end{array}\right]$ (2.3)
 > $\mathrm{CM4}≔\mathrm{Matrix}\left(\left[\left[2,-2,0,0,-1,0\right],\left[-1,2,0,0,0,0\right],\left[0,0,2,-1,0,0\right],\left[0,0,-1,2,0,-1\right],\left[-1,0,0,0,2,-1\right],\left[0,0,0,-1,-1,2\right]\right]\right)$
 ${\mathrm{CM4}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{2}& {0}& {0}& {-}{1}& {0}\\ {-}{1}& {2}& {0}& {0}& {0}& {0}\\ {0}& {0}& {2}& {-}{1}& {0}& {0}\\ {0}& {0}& {-}{1}& {2}& {0}& {-}{1}\\ {-}{1}& {0}& {0}& {0}& {2}& {-}{1}\\ {0}& {0}& {0}& {-}{1}& {-}{1}& {2}\end{array}\right]$ (2.4)

Here are the standard forms, permutation matrices and root types.

 > $\mathrm{C1},\mathrm{P1},\mathrm{T1}≔\mathrm{CartanMatrixToStandardForm}\left(\mathrm{CM1}\right)$
 ${\mathrm{C1}}{,}{\mathrm{P1}}{,}{\mathrm{T1}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}& {0}\\ {0}& {0}& {-}{1}& {2}& {-}{1}& {-}{1}\\ {0}& {0}& {0}& {-}{1}& {2}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {2}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]{,}{"D"}$ (2.5)
 > $\mathrm{C2},\mathrm{P2},\mathrm{T2}≔\mathrm{CartanMatrixToStandardForm}\left(\mathrm{CM2}\right)$
 ${\mathrm{C2}}{,}{\mathrm{P2}}{,}{\mathrm{T2}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}& {-}{1}\\ {0}& {0}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {2}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}\end{array}\right]{,}{"E"}$ (2.6)
 > $\mathrm{C3},\mathrm{P3},\mathrm{T3}≔\mathrm{CartanMatrixToStandardForm}\left(\mathrm{CM3}\right)$
 ${\mathrm{C3}}{,}{\mathrm{P3}}{,}{\mathrm{T3}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}& {0}\\ {0}& {0}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {-}{1}\\ {0}& {0}& {0}& {0}& {-}{2}& {2}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}{"C"}$ (2.7)
 > $\mathrm{C4},\mathrm{P4},\mathrm{T4}≔\mathrm{CartanMatrixToStandardForm}\left(\mathrm{CM4}\right)$
 ${\mathrm{C4}}{,}{\mathrm{P4}}{,}{\mathrm{T4}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}& {0}\\ {0}& {0}& {-}{1}& {2}& {-}{1}& {0}\\ {0}& {0}& {0}& {-}{1}& {2}& {-}{2}\\ {0}& {0}& {0}& {0}& {-}{1}& {2}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\end{array}\right]{,}{"B"}$ (2.8)
 alg > $\mathrm{C1},\mathrm{P1},\mathrm{T1}≔\mathrm{CartanMatrixToStandardForm}\left(\mathrm{CM1}\right)$
 ${\mathrm{C1}}{,}{\mathrm{P1}}{,}{\mathrm{T1}}{:=}\left[\begin{array}{rrrrrr}{2}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {2}& {-}{1}& {0}& {0}& {0}\\ {0}& {-}{1}& {2}& {-}{1}& {0}& {0}\\ {0}& {0}& {-}{1}& {2}& {-}{1}& {-}{1}\\ {0}& {0}& {0}& {-}{1}& {2}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {2}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]{,}{"D"}$ (2.9)

For each example the second output is a permutation matrix which transforms the given input Cartan matrix to its standard form.

 > $\mathrm{LinearAlgebra}:-\mathrm{Equal}\left({\mathrm{P1}}^{-1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{CM1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{P1},\mathrm{C1}\right)$
 ${\mathrm{true}}$ (2.10)
 > $\mathrm{LinearAlgebra}:-\mathrm{Equal}\left({\mathrm{P2}}^{-1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{CM2}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{P2},\mathrm{C2}\right)$
 ${\mathrm{true}}$ (2.11)
 > $\mathrm{LinearAlgebra}:-\mathrm{Equal}\left({\mathrm{P3}}^{-1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{CM3}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{P3},\mathrm{C3}\right)$
 ${\mathrm{true}}$ (2.12)
 > $\mathrm{LinearAlgebra}:-\mathrm{Equal}\left({\mathrm{P4}}^{-1}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{CM4}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{P4},\mathrm{C4}\right)$
 ${\mathrm{true}}$ (2.13)

Example 2.

We define a 21-dimensional simple Lie algebra and calculate its root type.

 > $\mathrm{LD}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{alg},\left[21\right]\right],\left[\left[\left[1,2,3\right],1\right],\left[\left[1,3,2\right],-1\right],\left[\left[1,4,5\right],1\right],\left[\left[1,5,4\right],-2\right],\left[\left[1,5,7\right],2\right],\left[\left[1,6,8\right],1\right],\left[\left[1,7,5\right],-1\right],\left[\left[1,8,6\right],-1\right],\left[\left[1,10,11\right],1\right],\left[\left[1,11,10\right],-2\right],\left[\left[1,11,13\right],2\right],\left[\left[1,12,14\right],1\right],\left[\left[1,13,11\right],-1\right],\left[\left[1,14,12\right],-1\right],\left[\left[1,16,17\right],1\right],\left[\left[1,17,16\right],-2\right],\left[\left[1,17,19\right],2\right],\left[\left[1,18,20\right],1\right],\left[\left[1,19,17\right],-1\right],\left[\left[1,20,18\right],-1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,6\right],1\right],\left[\left[2,5,8\right],1\right],\left[\left[2,6,4\right],-2\right],\left[\left[2,6,9\right],2\right],\left[\left[2,8,5\right],-1\right],\left[\left[2,9,6\right],-1\right],\left[\left[2,10,12\right],1\right],\left[\left[2,11,14\right],1\right],\left[\left[2,12,10\right],-2\right],\left[\left[2,12,15\right],2\right],\left[\left[2,14,11\right],-1\right],\left[\left[2,15,12\right],-1\right],\left[\left[2,16,18\right],1\right],\left[\left[2,17,20\right],1\right],\left[\left[2,18,16\right],-2\right],\left[\left[2,18,21\right],2\right],\left[\left[2,20,17\right],-1\right],\left[\left[2,21,18\right],-1\right],\left[\left[3,5,6\right],1\right],\left[\left[3,6,5\right],-1\right],\left[\left[3,7,8\right],1\right],\left[\left[3,8,7\right],-2\right],\left[\left[3,8,9\right],2\right],\left[\left[3,9,8\right],-1\right],\left[\left[3,11,12\right],1\right],\left[\left[3,12,11\right],-1\right],\left[\left[3,13,14\right],1\right],\left[\left[3,14,13\right],-2\right],\left[\left[3,14,15\right],2\right],\left[\left[3,15,14\right],-1\right],\left[\left[3,17,18\right],1\right],\left[\left[3,18,17\right],-1\right],\left[\left[3,19,20\right],1\right],\left[\left[3,20,19\right],-2\right],\left[\left[3,20,21\right],2\right],\left[\left[3,21,20\right],-1\right],\left[\left[4,5,1\right],1\right],\left[\left[4,6,2\right],1\right],\left[\left[4,10,16\right],2\right],\left[\left[4,11,17\right],1\right],\left[\left[4,12,18\right],1\right],\left[\left[4,16,10\right],-2\right],\left[\left[4,17,11\right],-1\right],\left[\left[4,18,12\right],-1\right],\left[\left[5,6,3\right],1\right],\left[\left[5,7,1\right],1\right],\left[\left[5,8,2\right],1\right],\left[\left[5,10,17\right],1\right],\left[\left[5,11,16\right],2\right],\left[\left[5,11,19\right],2\right],\left[\left[5,12,20\right],1\right],\left[\left[5,13,17\right],1\right],\left[\left[5,14,18\right],1\right],\left[\left[5,16,11\right],-1\right],\left[\left[5,17,10\right],-2\right],\left[\left[5,17,13\right],-2\right],\left[\left[5,18,14\right],-1\right],\left[\left[5,19,11\right],-1\right],\left[\left[5,20,12\right],-1\right],\left[\left[6,8,1\right],1\right],\left[\left[6,9,2\right],1\right],\left[\left[6,10,18\right],1\right],\left[\left[6,11,20\right],1\right],\left[\left[6,12,16\right],2\right],\left[\left[6,12,21\right],2\right],\left[\left[6,14,17\right],1\right],\left[\left[6,15,18\right],1\right],\left[\left[6,16,12\right],-1\right],\left[\left[6,17,14\right],-1\right],\left[\left[6,18,10\right],-2\right],\left[\left[6,18,15\right],-2\right],\left[\left[6,20,11\right],-1\right],\left[\left[6,21,12\right],-1\right],\left[\left[7,8,3\right],1\right],\left[\left[7,11,17\right],1\right],\left[\left[7,13,19\right],2\right],\left[\left[7,14,20\right],1\right],\left[\left[7,17,11\right],-1\right],\left[\left[7,19,13\right],-2\right],\left[\left[7,20,14\right],-1\right],\left[\left[8,9,3\right],1\right],\left[\left[8,11,18\right],1\right],\left[\left[8,12,17\right],1\right],\left[\left[8,13,20\right],1\right],\left[\left[8,14,19\right],2\right],\left[\left[8,14,21\right],2\right],\left[\left[8,15,20\right],1\right],\left[\left[8,17,12\right],-1\right],\left[\left[8,18,11\right],-1\right],\left[\left[8,19,14\right],-1\right],\left[\left[8,20,13\right],-2\right],\left[\left[8,20,15\right],-2\right],\left[\left[8,21,14\right],-1\right],\left[\left[9,12,18\right],1\right],\left[\left[9,14,20\right],1\right],\left[\left[9,15,21\right],2\right],\left[\left[9,18,12\right],-1\right],\left[\left[9,20,14\right],-1\right],\left[\left[9,21,15\right],-2\right],\left[\left[10,11,1\right],1\right],\left[\left[10,12,2\right],1\right],\left[\left[10,16,4\right],2\right],\left[\left[10,17,5\right],1\right],\left[\left[10,18,6\right],1\right],\left[\left[11,12,3\right],1\right],\left[\left[11,13,1\right],1\right],\left[\left[11,14,2\right],1\right],\left[\left[11,16,5\right],1\right],\left[\left[11,17,4\right],2\right],\left[\left[11,17,7\right],2\right],\left[\left[11,18,8\right],1\right],\left[\left[11,19,5\right],1\right],\left[\left[11,20,6\right],1\right],\left[\left[12,14,1\right],1\right],\left[\left[12,15,2\right],1\right],\left[\left[12,16,6\right],1\right],\left[\left[12,17,8\right],1\right],\left[\left[12,18,4\right],2\right],\left[\left[12,18,9\right],2\right],\left[\left[12,20,5\right],1\right],\left[\left[12,21,6\right],1\right],\left[\left[13,14,3\right],1\right],\left[\left[13,17,5\right],1\right],\left[\left[13,19,7\right],2\right],\left[\left[13,20,8\right],1\right],\left[\left[14,15,3\right],1\right],\left[\left[14,17,6\right],1\right],\left[\left[14,18,5\right],1\right],\left[\left[14,19,8\right],1\right],\left[\left[14,20,7\right],2\right],\left[\left[14,20,9\right],2\right],\left[\left[14,21,8\right],1\right],\left[\left[15,18,6\right],1\right],\left[\left[15,20,8\right],1\right],\left[\left[15,21,9\right],2\right],\left[\left[16,17,1\right],1\right],\left[\left[16,18,2\right],1\right],\left[\left[17,18,3\right],1\right],\left[\left[17,19,1\right],1\right],\left[\left[17,20,2\right],1\right],\left[\left[18,20,1\right],1\right],\left[\left[18,21,2\right],1\right],\left[\left[19,20,3\right],1\right],\left[\left[20,21,3\right],1\right]\right]\right]\right):$

Initialize this Lie algebra.

 > $\mathrm{DGsetup}\left(\mathrm{LD}\right):$

Find a Cartan subalgebra.

 alg > $\mathrm{CSA}≔\mathrm{CartanSubalgebra}\left(\right)$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e16}}{+}{\mathrm{e19}}{,}{\mathrm{e21}}\right]$ (2.14)

Find the root space decomposition.

 alg > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{0}{,}{-}{2}{}{I}\right]{=}{\mathrm{e9}}{+}{I}{}{\mathrm{e15}}{,}\left[{-}{2}{}{I}{,}{2}{}{I}{,}{0}\right]{=}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{-}{\mathrm{e7}}{-}{I}{}{\mathrm{e10}}{+}{\mathrm{e11}}{+}{I}{}{\mathrm{e13}}{,}\left[{I}{,}{I}{,}{I}\right]{=}{\mathrm{e6}}{-}{I}{}{\mathrm{e8}}{-}{I}{}{\mathrm{e12}}{-}{\mathrm{e14}}{,}\left[{-}{I}{,}{-}{I}{,}{-}{I}\right]{=}{\mathrm{e6}}{+}{I}{}{\mathrm{e8}}{+}{I}{}{\mathrm{e12}}{-}{\mathrm{e14}}{,}\left[{I}{,}{-}{I}{,}{I}\right]{=}{\mathrm{e2}}{-}{I}{}{\mathrm{e3}}{-}{I}{}{\mathrm{e18}}{-}{\mathrm{e20}}{,}\left[{-}{I}{,}{I}{,}{-}{I}\right]{=}{\mathrm{e2}}{+}{I}{}{\mathrm{e3}}{+}{I}{}{\mathrm{e18}}{-}{\mathrm{e20}}{,}\left[{2}{}{I}{,}{2}{}{I}{,}{0}\right]{=}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{-}{\mathrm{e7}}{-}{I}{}{\mathrm{e10}}{-}{\mathrm{e11}}{+}{I}{}{\mathrm{e13}}{,}\left[{2}{}{I}{,}{-}{2}{}{I}{,}{0}\right]{=}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{-}{\mathrm{e7}}{+}{I}{}{\mathrm{e10}}{+}{\mathrm{e11}}{-}{I}{}{\mathrm{e13}}{,}\left[{-}{2}{}{I}{,}{0}{,}{0}\right]{=}{\mathrm{e16}}{+}{I}{}{\mathrm{e17}}{-}{\mathrm{e19}}{,}\left[{I}{,}{I}{,}{-}{I}\right]{=}{\mathrm{e2}}{-}{I}{}{\mathrm{e3}}{+}{I}{}{\mathrm{e18}}{+}{\mathrm{e20}}{,}\left[{0}{,}{0}{,}{2}{}{I}\right]{=}{\mathrm{e9}}{-}{I}{}{\mathrm{e15}}{,}\left[{-}{I}{,}{-}{I}{,}{I}\right]{=}{\mathrm{e2}}{+}{I}{}{\mathrm{e3}}{-}{I}{}{\mathrm{e18}}{+}{\mathrm{e20}}{,}\left[{I}{,}{-}{I}{,}{-}{I}\right]{=}{\mathrm{e6}}{-}{I}{}{\mathrm{e8}}{+}{I}{}{\mathrm{e12}}{+}{\mathrm{e14}}{,}\left[{2}{}{I}{,}{0}{,}{0}\right]{=}{\mathrm{e16}}{-}{I}{}{\mathrm{e17}}{-}{\mathrm{e19}}{,}\left[{0}{,}{-}{2}{}{I}{,}{0}\right]{=}{\mathrm{e4}}{+}{\mathrm{e7}}{+}{I}{}{\mathrm{e10}}{+}{I}{}{\mathrm{e13}}{,}\left[{-}{I}{,}{I}{,}{I}\right]{=}{\mathrm{e6}}{+}{I}{}{\mathrm{e8}}{-}{I}{}{\mathrm{e12}}{+}{\mathrm{e14}}{,}\left[{0}{,}{2}{}{I}{,}{0}\right]{=}{\mathrm{e4}}{+}{\mathrm{e7}}{-}{I}{}{\mathrm{e10}}{-}{I}{}{\mathrm{e13}}{,}\left[{-}{2}{}{I}{,}{-}{2}{}{I}{,}{0}\right]{=}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{-}{\mathrm{e7}}{+}{I}{}{\mathrm{e10}}{-}{\mathrm{e11}}{-}{I}{}{\mathrm{e13}}\right]\right)$ (2.15)

Find the roots, positive roots and a choice of simple roots.

 alg > $\mathrm{RT}≔\mathrm{LieAlgebraRoots}\left(\mathrm{RSD}\right)$
 ${\mathrm{RT}}{:=}\left[\left[\begin{array}{c}{0}\\ {0}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {-}{2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {-}{2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {-}{2}{}{I}\\ {0}\end{array}\right]\right]$ (2.16)
 alg > $\mathrm{PR}≔\mathrm{PositiveRoots}\left(\mathrm{RT},⟨7I,3I,I⟩\right)$
 ${\mathrm{PR}}{:=}\left[\left[\begin{array}{c}{0}\\ {0}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {-}{2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {-}{2}{}{I}\\ {0}\end{array}\right]\right]$ (2.17)
 alg > $\mathrm{SR}≔\mathrm{SimpleRoots}\left(\mathrm{PR}\right)$

Find the Cartan matrix.

 alg > $\mathrm{CM}≔\mathrm{CartanMatrix}\left(\mathrm{SR},\mathrm{RSD}\right)$
 ${\mathrm{CM}}{:=}\left[\begin{array}{rrr}{2}& {0}& {-}{2}\\ {0}& {2}& {-}{1}\\ {-}{1}& {-}{1}& {2}\end{array}\right]$ (2.18)

Transform the Cartan matrix to standard form. Here we use the second calling sequence. The command CartanMatrixToStandardForm now returns a permuted set of simple roots for which the Cartan matrix will be in standard form.

 alg > $\mathrm{C1},\mathrm{S1},\mathrm{T1}≔\mathrm{CartanMatrixToStandardForm}\left(\mathrm{CM},\mathrm{SR}\right)$
 ${\mathrm{C1}}{,}{\mathrm{S1}}{,}{\mathrm{T1}}{:=}\left[\begin{array}{rrr}{2}& {-}{1}& {0}\\ {-}{1}& {2}& {-}{1}\\ {0}& {-}{2}& {2}\end{array}\right]{,}\left[\left[\begin{array}{c}{0}\\ {-}{2}{}{I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {-}{2}{}{I}\end{array}\right]\right]{,}{"C"}$ (2.19)

Check the result by re-calculating the Cartan matrix with respect to the permuted set of roots. We get the standard form immediately.

 alg > $\mathrm{CartanMatrix}\left(\mathrm{S1},\mathrm{RSD}\right)$
 $\left[\begin{array}{rrr}{2}& {-}{1}& {0}\\ {-}{1}& {2}& {-}{1}\\ {0}& {-}{2}& {2}\end{array}\right]$ (2.20)

The root type of our 21-dimensional Lie algebra is