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 involution of g is a Lie algebra automorphism Θ : g → g with ${\mathrm{\Θ}}^2\= \mathrm{Id}$ and such that the symmetric bilinear form $B_{{\mathrm{\Θ}}_{}}\left(x\,y\right) \= -⟨x\,{\mathrm{\Θ}}_{}\left(y\right)⟩$is positive-definite.

The command CartanInvolution returns a transformation defining a Cartan involution.

A Cartan decomposition is a vector space decomposition g = t ⊕ p , where t is a subalgebra, p a subspace, [t, p] ⊆ p , [p, p] ⊆ t and the Killing form is negative-definite on t and positive-definite on p.

Given a Cartan decomposition, the linear transformation which is the identity $\mathrm{Id}$on t and $-\mathrm{Id}$on p is a Cartan involution. This is the involution computed by the first calling sequence for the command CartanInvolution.

We remark that, conversely, given a Cartan involution$\mathrm{\Θ}$, the +1, -1 eigenspaces $E_{1 }\= \mathrm{\𝔭}$and $E_{-1}\= \mathrm{\𝔱}$yield a Cartan decomposition. Also, any two Cartan involutions ${\mathrm{\Θ}}_1$and ${\mathrm{\Θ}}_2$on g are related by an inner automorphism ${\mathrm{\φ}}_{} \: \mathrm{\𝔤} \to \mathrm{\𝔤}$, that is, ${\mathrm{\Θ}}_2\= \mathrm{\φ} {{\mathrm{\Theta }}_1}_{}$${\mathrm{\φ}}^{-1}$.

 A Cartan involution can also be calculated from a Cartan subalgebra, the associated root space decomposition and a choice of positive roots. The algorithm can be summarized as follows. First use the procedure Complexify to define the complexification ${\mathrm{\𝔤}}_C$ of the Lie algebra $\mathrm{\𝔤}$. This is a real semi-simple Lie algebra of twice the dimension of $\mathrm{\𝔤}$ . Let $\mathrm{\σ} \: {\mathrm{\𝔤}}_C \to {\mathrm{\𝔤}}_C$ denote the standard conjugation map. Next use the command SplitAndCompactForms to find a complex basis of $\mathrm{\𝔤}\mathit{ }$which defines a compact form$\mathrm{\𝔲}\mathit{ }$of $\mathrm{\𝔤}$. Identify $\mathrm{\𝔲}\mathit{ }$with a subalgebra of ${𝔤}_C$ and let $\mathrm{\τ}$be the corresponding conjugate map. One proves that $\mathrm{\τ}$is a Cartan involution of ${𝔤}_C$ . If $\mathrm{\τ}$restricts to a mapping $\mathrm{\τ}\: \mathrm{\𝔤}\mathit{ }\mathit{\to }\mathrm{\𝔤}$, then $\mathrm{\τ}$would be the required Cartan involution for $\mathrm{\𝔤}$. However, this generally is not the case so the idea to conjugate $\mathrm{\τ}$to another Cartan involution which does restrict to$𝔤$. Note that the requirement that $\mathrm{\τ}$restricts to a mapping $\mathrm{\τ}\: \mathrm{\𝔤}\mathit{ }\mathit{\to }\mathrm{\𝔤}\mathit{ }$ is equivalent to the requirement that $\mathrm{\τ}$commutes with $\mathrm{\σ}$. One proves that $\mathrm{\ψ} \= \mathrm{\σ} \mathrm{\τ} \mathrm{\σ} \mathrm{\τ}$is a linear transformation with positive eigenvalues. The required Cartan involution is then $\mathrm{\Θ}\={\mathrm{\ψ}}^{1\/4}\mathrm{\τ} {\mathrm{\ψ}}^{-1\/4}$. See A.Cap and J. Slovak, Parabolic Geometries I - Background and General Theory, page 203 for further details.

with(DifferentialGeometry): with(LieAlgebras):

LD := SimpleLieAlgebraData("so(3, 2)", so32, labelformat = "gl", labels = ['E', 'omega']):

DGsetup(LD);

$\mathrm{Lie\; algebra:\; so32}$

M := StandardRepresentation(so32);

T, P := CartanDecomposition(M, so32);

$T\,P:=\left[\mathrm{E12}-\mathrm{E21}\,\mathrm{E14}\+\mathrm{E32}\,\mathrm{E15}\+\mathrm{E35}\,\mathrm{E25}\+\mathrm{E45}\right]\,\left[\mathrm{E11}\,\mathrm{E12}\+\mathrm{E21}\,\mathrm{E22}\,\mathrm{E14}-\mathrm{E32}\,\mathrm{E15}-\mathrm{E35}\,\mathrm{E25}-\mathrm{E45}\right]$

Theta1 := CartanInvolution(T, P);

$\mathrm{\Θ1}:=\left[\left[\mathrm{E11}\,-\mathrm{E11}\right]\,\left[\mathrm{E12}\,-\mathrm{E21}\right]\,\left[\mathrm{E21}\,-\mathrm{E12}\right]\,\left[\mathrm{E22}\,-\mathrm{E22}\right]\,\left[\mathrm{E14}\,\mathrm{E32}\right]\,\left[\mathrm{E32}\,\mathrm{E14}\right]\,\left[\mathrm{E15}\,\mathrm{E35}\right]\,\left[\mathrm{E25}\,\mathrm{E45}\right]\,\left[\mathrm{E35}\,\mathrm{E15}\right]\,\left[\mathrm{E45}\,\mathrm{E25}\right]\right]$

ComposeTransformations(Theta1, Theta1);

$\left[\left[\mathrm{E11}\,\mathrm{E11}\right]\,\left[\mathrm{E12}\,\mathrm{E12}\right]\,\left[\mathrm{E21}\,\mathrm{E21}\right]\,\left[\mathrm{E22}\,\mathrm{E22}\right]\,\left[\mathrm{E14}\,\mathrm{E14}\right]\,\left[\mathrm{E32}\,\mathrm{E32}\right]\,\left[\mathrm{E15}\,\mathrm{E15}\right]\,\left[\mathrm{E25}\,\mathrm{E25}\right]\,\left[\mathrm{E35}\,\mathrm{E35}\right]\,\left[\mathrm{E45}\,\mathrm{E45}\right]\right]$

Query(Theta1, "Homomorphism");

$\mathrm{true}$

V := Tools:-DGinfo(so32, "FrameBaseVectors");

$V:=\left[\mathrm{E11}\,\mathrm{E12}\,\mathrm{E21}\,\mathrm{E22}\,\mathrm{E14}\,\mathrm{E32}\,\mathrm{E15}\,\mathrm{E25}\,\mathrm{E35}\,\mathrm{E45}\right]$

B := Matrix(10, 10, (i,j) -> Killing(-V[i], ApplyHomomorphism(Theta1, V[j])));

Query(Theta1, "CartanInvolution");

$\mathrm{true}$

CSA := CartanSubalgebra();

$\mathrm{CSA}:=\left[\mathrm{E11}\,\mathrm{E22}\right]$

RSD := RootSpaceDecomposition(CSA);

$\mathrm{RSD}:=\mathrm{table}\left(\left[\left[-1\,0\right]\=\mathrm{E35}\,\left[0\,-1\right]\=\mathrm{E45}\,\left[-1\,-1\right]\=\mathrm{E32}\,\left[1\,1\right]\=\mathrm{E14}\,\left[-1\,1\right]\=\mathrm{E21}\,\left[1\,-1\right]\=\mathrm{E12}\,\left[1\,0\right]\=\mathrm{E15}\,\left[0\,1\right]\=\mathrm{E25}\right]\right)$

PosRts := PositiveRoots(RSD);

Theta2 := CartanInvolution(CSA, RSD, PosRts);

$\mathrm{\Θ2}:=\left[\left[\mathrm{E11}\,-\mathrm{E11}\right]\,\left[\mathrm{E12}\,-\mathrm{E21}\right]\,\left[\mathrm{E21}\,-\mathrm{E12}\right]\,\left[\mathrm{E22}\,-\mathrm{E22}\right]\,\left[\mathrm{E14}\,\frac{1}{4}\mathrm{E32}\right]\,\left[\mathrm{E32}\,4\mathrm{E14}\right]\,\left[\mathrm{E15}\,\frac{1}{2}\mathrm{E35}\right]\,\left[\mathrm{E25}\,\frac{1}{2}\mathrm{E45}\right]\,\left[\mathrm{E35}\,2\mathrm{E15}\right]\,\left[\mathrm{E45}\,2\mathrm{E25}\right]\right]$

Query(Theta2, "CartanInvolution");

$\mathrm{true}$

Theta1;

$\left[\left[\mathrm{E11}\,-\mathrm{E11}\right]\,\left[\mathrm{E12}\,-\mathrm{E21}\right]\,\left[\mathrm{E21}\,-\mathrm{E12}\right]\,\left[\mathrm{E22}\,-\mathrm{E22}\right]\,\left[\mathrm{E14}\,\mathrm{E32}\right]\,\left[\mathrm{E32}\,\mathrm{E14}\right]\,\left[\mathrm{E15}\,\mathrm{E35}\right]\,\left[\mathrm{E25}\,\mathrm{E45}\right]\,\left[\mathrm{E35}\,\mathrm{E15}\right]\,\left[\mathrm{E45}\,\mathrm{E25}\right]\right]$

A := AdjointExp(evalDG(2*E35));

phi := Transformation(so32, so32, A);

$\mathrm{\φ}:=\left[\left[\mathrm{E11}\,\mathrm{E11}\+2\mathrm{E35}\right]\,\left[\mathrm{E12}\,\mathrm{E12}\+2\mathrm{E32}\+2\mathrm{E45}\right]\,\left[\mathrm{E21}\,\mathrm{E21}\right]\,\left[\mathrm{E22}\,\mathrm{E22}\right]\,\left[\mathrm{E14}\,-2\mathrm{E21}\+\mathrm{E14}-2\mathrm{E25}\right]\,\left[\mathrm{E32}\,\mathrm{E32}\right]\,\left[\mathrm{E15}\,2\mathrm{E11}\+\mathrm{E15}\+2\mathrm{E35}\right]\,\left[\mathrm{E25}\,2\mathrm{E21}\+\mathrm{E25}\right]\,\left[\mathrm{E35}\,\mathrm{E35}\right]\,\left[\mathrm{E45}\,2\mathrm{E32}\+\mathrm{E45}\right]\right]$

newTheta := ComposeTransformations(phi, Theta1, InverseTransformation(phi));

$\mathrm{newTheta}:=\left[\left[\mathrm{E11}\,-5\mathrm{E11}-2\mathrm{E15}-6\mathrm{E35}\right]\,\left[\mathrm{E12}\,-9\mathrm{E21}\+2\mathrm{E14}-6\mathrm{E25}\right]\,\left[\mathrm{E21}\,-\mathrm{E12}-2\mathrm{E32}-2\mathrm{E45}\right]\,\left[\mathrm{E22}\,-\mathrm{E22}\right]\,\left[\mathrm{E14}\,2\mathrm{E12}\+9\mathrm{E32}\+6\mathrm{E45}\right]\,\left[\mathrm{E32}\,-2\mathrm{E21}\+\mathrm{E14}-2\mathrm{E25}\right]\,\left[\mathrm{E15}\,6\mathrm{E11}\+2\mathrm{E15}\+9\mathrm{E35}\right]\,\left[\mathrm{E25}\,2\mathrm{E12}\+6\mathrm{E32}\+5\mathrm{E45}\right]\,\left[\mathrm{E35}\,2\mathrm{E11}\+\mathrm{E15}\+2\mathrm{E35}\right]\,\left[\mathrm{E45}\,6\mathrm{E21}-2\mathrm{E14}\+5\mathrm{E25}\right]\right]$

Query(newTheta, "CartanInvolution");

$\mathrm{true}$