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Query[CartanInvolution] - check if a linear transformation of a semi-simple, real Lie algebra is a Cartan involution
Calling Sequences
Query(
Parameters
 Theta - a transformation, mapping a semi-simple Lie algebra to itself
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Description
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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.
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A Cartan involution of g is a Lie algebra automorphism Θ : g → g such that [i] , and [ii] the symmetricbilinear form  = -`<,>`(x, Theta(y))](/support/helpjp/helpview.aspx?si=7181/file07644/math37.png) is positive-definite.
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Examples
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We check to see if some transformations of  are Cartan involutions. Initialize the Lie algebra 
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![LD := LieAlgebraData([[h, x] = 2*x, [h, y] = -2*y, [x, y] = h], [h, x, y], sl2)](/support/helpjp/helpview.aspx?si=7181/file07644/math60.png)
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![_DG([["LieAlgebra", sl2, [3, table( [ ] )]], [[[1, 2, 2], 2], [[1, 3, 3], -2], [[2, 3, 1], 1]]])](/support/helpjp/helpview.aspx?si=7181/file07644/math63.png)
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Define a transformation ![Theta[1]](/support/helpjp/helpview.aspx?si=7181/file07644/math75.png) and check that it is an involution.
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sl2 >
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![Theta1 := Transformation([[e1, -e1], [e2, -e3], [e3, -e2]])](/support/helpjp/helpview.aspx?si=7181/file07644/math80.png)
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![_DG([["transformation", [[sl2, 0], [sl2, 0]], [], [Matrix(%id = 87970140)]], [[-_z1, _z1], [-_z3, _z2], [-_z2, _z3]]])](/support/helpjp/helpview.aspx?si=7181/file07644/math83.png)
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sl2 >
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Define a transformation ![Theta[2].](/support/helpjp/helpview.aspx?si=7181/file07644/math95.png) It is a homomorphism,  , but the symmetric bilinear form  = -`<,>`(x, Theta(y))](/support/helpjp/helpview.aspx?si=7181/file07644/math99.png) is not positive-definite.
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sl2 >
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![Theta2 := Transformation([[e1, e1], [e2, -e2], [e3, -e3]])](/support/helpjp/helpview.aspx?si=7181/file07644/math104.png)
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![_DG([["transformation", [[sl2, 0], [sl2, 0]], [], [Matrix(%id = 91080784)]], [[_z1, _z1], [-_z2, _z2], [-_z3, _z3]]])](/support/helpjp/helpview.aspx?si=7181/file07644/math107.png)
| (2.5) |
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sl2 >
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| (2.6) |
The map ![Theta[2]](/support/helpjp/helpview.aspx?si=7181/file07644/math119.png) is a homomorphism.
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sl2 >
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| (2.7) |
The map ![Theta[2]](/support/helpjp/helpview.aspx?si=7181/file07644/math132.png) satisfies ![Theta[2]^2 = Id](/support/helpjp/helpview.aspx?si=7181/file07644/math134.png) ,
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![_DG([["transformation", [[sl2, 0], [sl2, 0]], [], [Matrix(%id = 93880624)]], [[_z1, _z1], [_z2, _z2], [_z3, _z3]]])](/support/helpjp/helpview.aspx?si=7181/file07644/math142.png)
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The symmetric bilinear form  = -`<,>`(x, Theta(y))](/support/helpjp/helpview.aspx?si=7181/file07644/math149.png) is not positive-definite.
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![V := [e1, e2, e3]](/support/helpjp/helpview.aspx?si=7181/file07644/math154.png)
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![[_DG([["vector", sl2, []], [[[1], 1]]]), _DG([["vector", sl2, []], [[[2], 1]]]), _DG([["vector", sl2, []], [[[3], 1]]])]](/support/helpjp/helpview.aspx?si=7181/file07644/math157.png)
| (2.9) |
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![Matrix(3, 3, proc (i, j) options operator, arrow; Killing(V[i], ApplyHomomorphism(Theta2, V[j])) end proc)](/support/helpjp/helpview.aspx?si=7181/file07644/math161.png)
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| (2.10) |
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