Elementary property of maximal compact subgroup

[quasar987]

It is said on wiki* that

"Maximal compact subgroups are not unique unless the group G is a semidirect product of a compact group and a contractible group, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g in G such that $gKg^{-1}=L$ – hence a maximal compact subgroup is essentially unique, and people often speak of "the" maximal compact subgroup."

Why is that so? If the action of G on itself by conjugation were transitive it would be obvious but it isn't, is it?


*http://en.wikipedia.org/wiki/Maximal_compact_subgroup#Existence_and_uniqueness

 

[quasar987]

Just to clarify, what I am asking about is the "unique up to conjugation" part, not the first part about uniqueness in the case G is a semidirect product of a compact group and a contractible group.