== (A) complex conjugation on S^3
J0^2 = -I: True
e1, e2 orthogonal to p and J0 p (so they span H_p): [True, True, True, True]
|e1|^2 = |e2|^2 = |p|^2: True True
sigma orthogonal (S^T S = I), so sigma preserves S^3 and the Euclidean metric: True
sigma anti-commutes with J0 (S J0 = -J0 S): True
d sigma maps H_p into H_sigma(p): [True, True, True, True]
sigma^* theta = -theta: True
d theta(e1, J0 e1) = 2|p|^2 (> 0 on S^3, theta adapted): True
theta = <J0 p, dp> has d theta = 2(dx1^dy1+dx2^dy2): coefficients [-1, 1] (d(-y1 dx1 + x1 dy1) = 2 dx1^dy1)
== (B) (x,y,t) -> (x,-y,-t) on the Heisenberg group
theta(X) = theta(Y) = 0: True True
d theta(X, JX) = d theta(X, Y) = 4 > 0 (d theta = 4 dx^dy): True
d tau X = X at tau(p): True
d tau Y = -Y at tau(p): True
tau^* theta = -theta: True
Hence d tau J = -J d tau on H, and d tau maps the orthonormal frame (X, Y) to (X, -Y): an isometry.
