Geodesic
Conjugacy of polar factorizations of Lie groups
Sbornik. Mathematics, Tome 13 (1971) no. 1, pp. 12-24

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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. Bibliography: 5 titles. 
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D. V. Alekseevskii. Conjugacy of polar factorizations of Lie groups. Sbornik. Mathematics, Tome 13 (1971) no. 1, pp. 12-24. http://geodesic.mathdoc.fr/item/SM_1971_13_1_a1/