 CartanDecomposition - Maple Help

LieAlgebras[CartanDecomposition] - find a Cartan decomposition of a non-compact semi-simple Lie algebra

Calling Sequences

CartanDecomposition(${\mathbf{Θ}}$)

CartanDecomposition(A, alg)

CartanDecomposition(Alg, CSA, RSD, PosRts)

Parameters

$\mathrm{Θ}$       - a transformation, defining a Cartan involution of a non-compact, semi-simple real Lie algebra

A       - a list of square matrices, defining a Lie algebra and closed under Hermitian transposition

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

CSA     - a list of vectors, defining a Cartan subalgebra of a Lie algebra

RSD     - a table, specifying the root space decomposition of the Lie algebra with respect to the Cartan subalgebra CSA

PosRts  - a list of vectors, specifying a choice of positive roots for the root space decomposition Description

 • Let g be a semi-simple real Lie algebra. Then g is called compact if the Killing form of g is negative-definite, otherwise g is called non-compact.
 • A Cartan decomposition is a vector space decomposition g = t ⊕ p, where t is a subalgebra, p a subspace, [t, p] ⊆ p and [p, p] ⊆ t, the Killing form is negative-definite on t and positive-definite on p. The command CartanDecomposition returns 2 lists of vectors. The first list spans the subalgebra t and the second list spans the subspace p. The 3 different calling sequences for CartanDecomposition compute the Cartan decomposition from different data.
 • A Cartan involution of g is a Lie algebra automorphism Θ : g → g with and such that the symmetric bilinear form is positive-definite. Given a Cartan involution, the +1, -1 eigenspaces of $\mathrm{\Theta }$ yield a Cartan decomposition of g . This method of finding a Cartan decomposition is used by the first calling sequence CartanDecomposition(${\mathbf{Θ}}$).
 • For a semi-simple matrix algebra which is closed under Hermitian transposition, the decomposition into skew-Hermitian and Hermitian matrices will give a Cartan decomposition. This method of finding a Cartan decomposition is used by the second calling sequence CartanDecomposition(A, alg).
 • A Cartan decomposition may also be computed from a Cartan subalgebra, a root space decomposition, and a choice of positive roots. From these data a Cartan involution can be determined and the Cartan decomposition derived from it by the third calling sequence. Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We find a Cartan decomposition for the Lie algebra $\mathrm{sl}\left(3\right)$from a Cartan involution.

 > LD := SimpleLieAlgebraData("sl(3)", sl3, labelformat = "gl", labels = ['E', 'omega']):
 > DGsetup(LD);
 ${\mathrm{Lie algebra: sl3}}$ (2.1)

The transformation defines a Cartan involution for $\mathrm{sl}\left(3\right)$. With respect to the standard basis for $\mathrm{sl}\left(3\right)$ in terms of elementary matrices, this transformation is given by

 sl3 > Theta := Transformation([[E11, -E11], [E22, -E22], [E12, -E21], [E13, -E31], [E21, -E12], [E23, -E32], [E31, -E13], [E32, -E23]]);
 ${\mathrm{Θ}}{:=}\left[\left[{\mathrm{E11}}{,}{-}{\mathrm{E11}}\right]{,}\left[{\mathrm{E22}}{,}{-}{\mathrm{E22}}\right]{,}\left[{\mathrm{E12}}{,}{-}{\mathrm{E21}}\right]{,}\left[{\mathrm{E13}}{,}{-}{\mathrm{E31}}\right]{,}\left[{\mathrm{E21}}{,}{-}{\mathrm{E12}}\right]{,}\left[{\mathrm{E23}}{,}{-}{\mathrm{E32}}\right]{,}\left[{\mathrm{E31}}{,}{-}{\mathrm{E13}}\right]{,}\left[{\mathrm{E32}}{,}{-}{\mathrm{E23}}\right]\right]$ (2.2)

The corresponding Cartan decomposition is given by

 sl3 > T, P := CartanDecomposition(Theta);
 ${T}{,}{P}{:=}\left[{\mathrm{E12}}{-}{\mathrm{E21}}{,}{\mathrm{E13}}{-}{\mathrm{E31}}{,}{\mathrm{E23}}{-}{\mathrm{E32}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{+}{\mathrm{E21}}{,}{\mathrm{E13}}{+}{\mathrm{E31}}{,}{\mathrm{E23}}{+}{\mathrm{E32}}\right]$ (2.3)

Let us check the various properties of this decomposition.

1. $T$ is a subalgebra.

 sl3 > Query(T, "Subalgebra");
 ${\mathrm{true}}$ (2.4)

2. [

 sl3 > A := BracketOfSubspaces(T, P);
 ${A}{:=}\left[{-}{\mathrm{E12}}{-}{\mathrm{E21}}{,}{2}{}{\mathrm{E11}}{-}{2}{}{\mathrm{E22}}{,}{-}{\mathrm{E23}}{-}{\mathrm{E32}}{,}{\mathrm{E13}}{+}{\mathrm{E31}}{,}{2}{}{\mathrm{E11}}\right]$ (2.5)
 sl3 > GetComponents(A, P, trueorfalse = "on");
 ${\mathrm{true}}$ (2.6)

3. [

 sl3 > B := BracketOfSubspaces(P,P);
 ${B}{:=}\left[{\mathrm{E12}}{-}{\mathrm{E21}}{,}{2}{}{\mathrm{E13}}{-}{2}{}{\mathrm{E31}}{,}{\mathrm{E23}}{-}{\mathrm{E32}}\right]$ (2.7)
 sl3 > GetComponents(B, T, trueorfalse = "on");
 ${\mathrm{true}}$ (2.8)

4. Equivalent to 1, 2 and 3 is the fact that form a symmetric pair.

 sl3 > Query(T, P, "SymmetricPair");
 ${\mathrm{true}}$ (2.9)

5. The Killing form is negative-definite on T.

 sl3 > Killing(T);
 $\left[\begin{array}{rrr}{-}{12}& {0}& {0}\\ {0}& {-}{12}& {0}\\ {0}& {0}& {-}{12}\end{array}\right]$ (2.10)

6. The Killing form is positive-definite on P.

 sl3 > KP := Killing(P);
 ${\mathrm{KP}}{:=}\left[\begin{array}{rrrrr}{12}& {6}& {0}& {0}& {0}\\ {6}& {12}& {0}& {0}& {0}\\ {0}& {0}& {12}& {0}& {0}\\ {0}& {0}& {0}& {12}& {0}\\ {0}& {0}& {0}& {0}& {12}\end{array}\right]$ (2.11)
 sl3 > LinearAlgebra:-IsDefinite(KP);
 ${\mathrm{true}}$ (2.12)

All of these properties of the Cartan decomposition can be checked at once with the Query/"CartanDecomposition" command.

 sl3 > Query(T, P, "CartanDecomposition");
 ${\mathrm{true}}$ (2.13)

Example 2.

Here we shall calculate the Cartan decomposition of $\mathrm{sl}\left(3\right)$ from its standard matrix representation. We use the Lie algebra initialized in Example 1. The command StandardRepresentation can be applied to any Lie algebra created by SimpleLieAlgebraData.

 sl3 > M := StandardRepresentation(sl3);
 ${M}{:=}\left[\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {1}& {0}\end{array}\right]\right]$ (2.14)

Calculate the Cartan decomposition using the matrices (2.14), returning the answer as vectors in the Lie algebra $\mathrm{sl}\left(3\right).$

 sl3 > T2, P2 := CartanDecomposition(M, sl3);
 ${\mathrm{T2}}{,}{\mathrm{P2}}{:=}\left[{\mathrm{E12}}{-}{\mathrm{E21}}{,}{\mathrm{E13}}{-}{\mathrm{E31}}{,}{\mathrm{E23}}{-}{\mathrm{E32}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E22}}{,}{\mathrm{E12}}{+}{\mathrm{E21}}{,}{\mathrm{E13}}{+}{\mathrm{E31}}{,}{\mathrm{E23}}{+}{\mathrm{E32}}\right]$ (2.15)

The Cartan decomposition in terms of matrices can be obtained as follows.

 sl3 > T3, P3 := CartanDecomposition(M);
 ${\mathrm{T3}}{,}{\mathrm{P3}}{:=}\left[\left[\begin{array}{rrr}{0}& {1}& {0}\\ {-}{1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {-}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {-}{1}& {0}\end{array}\right]\right]{,}\left[\left[\begin{array}{rrr}{2}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {-}{2}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {2}& {0}\\ {0}& {0}& {-}{2}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {1}& {0}\end{array}\right]\right]$ (2.16)

Note that the matrices in are skew-symmetric and that the matrices in are symmetric.

Example 3.

Here we shall calculate the Cartan decomposition of from its standard matrix representation.

 > LD := SimpleLieAlgebraData("sp(4, 2)", sp42, labelformat = "gl", labels = ['F', 'sigma']):
 > DGsetup(LD);
 ${\mathrm{Lie algebra: sp42}}$ (2.17)
 sl3 > M := StandardRepresentation(sp42);
 ${M}{:=}\left[\left[\begin{array}{rrrrrr}{0}& {-}{1}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{I}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{I}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {I}& {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{I}& {0}\\ {0}& {0}& {0}& {-}{I}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{I}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}& {-}{1}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {-}{I}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{I}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}& {0}& {0}\\ {0}& {-}{I}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}& {-}{I}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {-}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {I}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {I}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}& {0}& {0}\\ {0}& {0}& {-}{I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {-}{I}& {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{I}& {0}& {0}& {0}\\ {0}& {-}{I}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{I}\end{array}\right]{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccccc}{0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}& {0}& {0}\end{array}\right]\right]$ (2.18)

Here is a Cartan decomposition for $\mathrm{sp}\left(4,2\right)$.

 sp42 > T, P := CartanDecomposition(M, sp42);
 ${T}{,}{P}{:=}\left[{\mathrm{F12}}{,}{\mathrm{Fi11}}{,}{\mathrm{Fi12}}{,}{\mathrm{Fi22}}{,}{\mathrm{F14}}{,}{\mathrm{F15}}{,}{\mathrm{F25}}{,}{\mathrm{Fi14}}{,}{\mathrm{Fi15}}{,}{\mathrm{Fi25}}{,}{\mathrm{Fi33}}{,}{\mathrm{F36}}{,}{\mathrm{Fi36}}\right]{,}\left[{\mathrm{F13}}{,}{\mathrm{F23}}{,}{\mathrm{Fi13}}{,}{\mathrm{Fi23}}{,}{\mathrm{F16}}{,}{\mathrm{F26}}{,}{\mathrm{Fi16}}{,}{\mathrm{Fi26}}\right]$ (2.19)

Check it.

 sp42 > Query(T, P, "CartanDecomposition");
 ${\mathrm{true}}$ (2.20)

Example 4.

We use the third calling sequence to calculate the Cartan decomposition for the split real form of the exceptional Lie algebra ${g}_{2}$.

 sp42 > LD4 := SimpleLieAlgebraData("g(2, Split)", g2);
 ${\mathrm{LD4}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{3}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{3}{}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{3}{}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{3}{}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{-}{3}{}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e11}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{3}{}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{3}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e1}}{-}{3}{}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{3}{}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{3}{}{\mathrm{e13}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{-}{3}{}{\mathrm{e14}}\right]$ (2.21)
 > DGsetup(LD4);
 ${\mathrm{Lie algebra: g2}}$ (2.22)

Calculate a Cartan subalgebra, a root space decomposition and a choice of positive roots.

 sp42 > CSA := CartanSubalgebra();
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]$ (2.23)
 sp42 > RSD := RootSpaceDecomposition(CSA);
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{1}\right]{=}{\mathrm{e8}}{,}\left[{1}{,}{0}\right]{=}{\mathrm{e6}}{,}\left[{3}{,}{-}{2}\right]{=}{\mathrm{e10}}{,}\left[{0}{,}{-}{1}\right]{=}{\mathrm{e14}}{,}\left[{-}{3}{,}{1}\right]{=}{\mathrm{e13}}{,}\left[{-}{3}{,}{2}\right]{=}{\mathrm{e4}}{,}\left[{3}{,}{-}{1}\right]{=}{\mathrm{e7}}{,}\left[{-}{1}{,}{1}\right]{=}{\mathrm{e5}}{,}\left[{-}{2}{,}{1}\right]{=}{\mathrm{e9}}{,}\left[{2}{,}{-}{1}\right]{=}{\mathrm{e3}}{,}\left[{1}{,}{-}{1}\right]{=}{\mathrm{e11}}{,}\left[{-}{1}{,}{0}\right]{=}{\mathrm{e12}}\right]\right)$ (2.24)
 g2 > PosRts := PositiveRoots(RSD);
 ${\mathrm{PosRts}}{:=}\left[\left[\begin{array}{r}{0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{2}\end{array}\right]{,}\left[\begin{array}{r}{3}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{2}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {-}{1}\end{array}\right]\right]$ (2.25)

Find the Cartan decomposition.

 g2 > T, P := CartanDecomposition(CSA, RSD, PosRts);
 ${T}{,}{P}{:=}\left[{\mathrm{e3}}{+}{\mathrm{e9}}{,}{\mathrm{e4}}{+}{\mathrm{e10}}{,}{\mathrm{e5}}{+}{\mathrm{e11}}{,}{\mathrm{e6}}{+}{\mathrm{e12}}{,}{\mathrm{e7}}{+}{\mathrm{e13}}{,}{\mathrm{e8}}{+}{\mathrm{e14}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{-}{\mathrm{e9}}{,}{\mathrm{e4}}{-}{\mathrm{e10}}{,}{\mathrm{e5}}{-}{\mathrm{e11}}{,}{\mathrm{e6}}{-}{\mathrm{e12}}{,}{\mathrm{e7}}{-}{\mathrm{e13}}{,}{\mathrm{e8}}{-}{\mathrm{e14}}\right]$ (2.26)
 g2 > Query(T, P, "CartanDecomposition");
 ${\mathrm{true}}$ (2.27)