The theorem of Lie and hyperplane subalgebras of Lie algebras (Q1194377)

Let \({\mathfrak g}\) be a finite dimensional Lie algebra over a field \(\Phi\). If the characteristic of \(\Phi\) is zero a theorem of Tits which goes back to Lie asserts that each hyperplane subalgebra \({\mathfrak h}\) of \({\mathfrak g}\) contains a codimension two ideal \({\mathfrak c}\) such that \({\mathfrak g}/{\mathfrak c}\) is either \(\Phi\), \(\text{sl}(2,\Phi)\) or the two dimensional non- abelian algebra \({\mathfrak s}_ 2\). A computational approach to this fact was given by \textit{K. H. Hofmann} in [Geom. Dedicata 36, 207-224 (1990; Zbl 0718.17006)]. There, it was left open if the intersection \(\Delta_{aff}({\mathfrak g})\) of all the \({\mathfrak c}\)'s for which \({\mathfrak g}/{\mathfrak c}\cong{\mathfrak s}_ 2\) is a characteristic ideal. In the paper under review an elementary proof for this fact is given. On the way the author presents a new proof of Lie's theorem and an alternative construction of \(\Delta_{aff}({\mathfrak g})\). Moreover he gives an example for the fact that \(\Delta_{aff}({\mathfrak g})\) is not a characteristic ideal if \(\Phi\) has finite characteristic.