Some structural results for nonsymmetric pairs of Lie algebras (Q1125917)

This paper deals with some structural results on pairs \((\mathfrak g,\mathfrak g_1)\) consisting of a Lie algebra \(\mathfrak g\) and a subalgebra \(\mathfrak g_1\) that is reductive in \(\mathfrak g\). First of all, if \(\mathfrak g_1\) is an absolutely simple subalgebra, a Killing orthogonal decomposition \(\mathfrak g=\mathfrak g_1\oplus{\mathfrak p}\) is established. It is analogous to the usual orthogonal decomposition for symmetric Lie algebra pairs except that \([{\mathfrak p},{\mathfrak p}]\) will not be contained in \(\mathfrak g_1\) unless \((\mathfrak g,\mathfrak g_1)\) arises from a symmetric Lie algebra. Secondly, for a general pair \((\mathfrak g,\mathfrak g_1)\) with \(\mathfrak g\) semisimple, a recipe is given for checking whether Cartan subalgebras of \(\mathfrak g_1\) can be uniquely extended to a Cartan subalgebra of \(\mathfrak g\). In addition the pair \((B_3,G_2)\) is carefully studied.