On the large subgroups of \(\mathrm{O}(n-1,1)\) and a result by Kowalsky and Witte (Q1938605)

The author considers Lie subgroups of two maximal dimensions in the Lorenz group \(\mathrm{SO}(n-1,1)\). In fact he gives a classification of corresponding Lie subalgebras in the Lie algebra \(\mathrm{so}(n-1,1)\). In 1998 he proved that for \(n >2\) and \(n \neq 5\), the largest possible dimension of a Lie subalgebra of \(\mathrm{so}(n-1,1)\) equals \(1/2(n-1)(n-2) +1\) and that subalgebras of such dimension are conjugate to some special subalgebra \(M(n)\) (which is described by its basis). Now he considers subalgebras of next dimension \(1/2(n-1)(n-2)\). It is necessary to remark that in the introduction and in the header of the first paragraph the author speaks about subalgebras of dimension \(1/2n(n-1)\), but in fact in this paper are considered subalgebras of dimension \(1/2(n-1)(n-2)\) (for which the main result of this paper is formulated). It is proved that for \(n \geq 5\) a \(1/2(n-1)(n-2)\)-dimensional Lie subalgebra of \(\mathrm{so}(n-1,1)\) is conjugated with one of the following three subalgebras: \(\mathrm{so}(n-1)\), \(\mathrm{so}(n-2,1)\), \(H(n)\). The subalgebras \(H(n)\) are described by their bases, the algebraic structure of them is not specified. It is noted that \(H(n)\) is a subalgebra in \(M(n)\), which in turn is a maximal subalgebra in the Lie algebra \(\mathrm{so}(n-1,1)\) (not in \(\mathrm{so}(n-2,1)\) as it is written in this paper, see page 62). Some geometric interpretations of this classification and a reformulation of some result of Kowalsky and Witte are given.