A related example is the circle group SO(2) inside SL(2, R).
The Cartan-Iwasawa-Malcev theorem asserts that every connected Lie group (and indeed every connected locally compact group) admits maximal compact subgroups and that they are all conjugate to one another.
For a semisimple Lie group uniqueness is a consequence of the Cartan fixed point theorem, which asserts that if a compact group acts by isometries on a complete simply connected negatively curved Riemannian manifold then it has a fixed point.
For a real semisimple Lie group, Cartan's proof of the existence and uniqueness of a maximal compact subgroup can be found in Borel (1950) and Helgason (1978).
For semisimple groups, existence is a consequence of the existence of a compact real form of the noncompact semisimple Lie group and the corresponding Cartan decomposition.
The proof of uniqueness relies on the fact that the corresponding Riemannian symmetric space G/K has negative curvature and Cartan's fixed point theorem.
Mostow (1955) showed that the derivative of the exponential map at any point of G/K satisfies |d exp X| ≥ |X|.
Uniqueness can then be deduced from the Bruhat-Tits fixed point theorem.
In particular a compact group acting by isometries must fix the circumcenter of each of its orbits.
Mostow (1955) also related the general problem for semisimple groups to the case of GL(n, R).
A direct proof of uniqueness relying on elementary properties of this space is given in Hilgert & Neeb (2012).
be a real semisimple Lie algebra with Cartan involution σ.
If H is another compact subgroup of G, then averaging the inner product over H with respect to the Haar measure gives an inner product invariant under H. The operators Ad p with p in P are positive symmetric operators.
is defined by the Cartan decomposition If fi is an orthonormal basis of eigenvectors of ad X with corresponding real eigenvalues μi, then Since the right hand side is a positive combination of exponentials, the real-valued function g is strictly convex if X ≠ 0, so has a unique minimum.
On the other hand, it has local minima at t = 0 and t = 1, hence X = 0 and p = exp Y is the unique global minimum.