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  and such that the symmetric bilinear form is positive-definite.

A Cartan decomposition is a vector space decomposition g = t 4p , where t is a subalgebra, p a subspace, [t, p] 4 p , [p, p] 4 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 on t and on p is a Cartan involution.  This is the involution computed by the command CartanInvolution.

Conversely, given a Cartan involution, the +1, -1 eigenspaces  and  yield a Cartan decomposition.

Any two Cartan involutions and  on g are related by an inner automorphism , that is, .

with(DifferentialGeometry): with(LieAlgebras):

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

DGsetup(LD);

M := StandardRepresentation(so32);

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

Theta := CartanInvolution(T, P);

ComposeTransformations(Theta, Theta);

Query(Theta, "Homomorphism");

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

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

Query(Theta, "CartanInvolution");

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

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

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

Query(newTheta, "CartanInvolution");