I'm studying the maximal compact subgroups of $PSL(2,\mathbb C)$.
According to Wikipedia, "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."
Can I deduce that $P S U(2,\mathbb C)$ is the only maximal conpact subgroup of $PSL(2,\mathbb C)$? And how can I show the "uniqueness" by the Cartan fixed point theorem?