4.42 (1973)

For what natural numbers $n$ does the following equality hold:
$$GL_n(\mathbb{R}) = D_n(\mathbb{R}) \cdot O_n(\mathbb{R}) \cdot UT_n(\mathbb{R}) \cdot GL_n(\mathbb{Z})?$$ For notation, see, for instance, (M. I. Kargapolov, Ju. I. Merzljakov, Fundamentals of the Theory of Groups, Springer, New York, 1979). For given $n$ this equality implies the affirmative solution of Minkowski’s problem on the product of $n$ linear forms (A. M. Macbeath, Proc. Glasgow Math. Assoc., 5, no. 2 (1961), 86–89), which remains open for $n \geqslant 6$. It is known that the equality holds for $n \leqslant 3$ (Kh. N. Narzullayev, Math. Notes, 18 (1975), 713–719); on the other hand, it does not hold for all sufficiently large $n$ (N. S. Akhmedov, Zapiski Nauchn. Seminarov LOMI, 67 (1977), 86–107 (Russian)).