Conjugacy of polar factorizations of Lie groups

A Lie group is said to be effective if it is connected and contains no compact normal divisors. A factorization of a connected Lie group into the product of two connected subgroups, the first of which is maximally compact and the second completely solvable is called a polar factorization.
In this article the following theorem is proved.
Theorem. Any two polar factorizations of an effective Lie group are conjugate under an inner automorphism.
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