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On Cartan subalgebras and Cartan subspaces of nonsymmetric pairs of Lie algebras

Let (g, g_1) be a pair of Lie algebras, defined over a field of characteristic zero, where g is semisimple and g_1 is a subalgebra reductive in g. We prove a result giving a necessary and sufficient technical condition so that the following holds: (Q1) For any Cartan subalgebra $h_1\subseteq g_1$ there exists a unique Cartan subalgebra $h\subseteq g$ containing h_1. Next we study a class of pairs (g, g_1), satisfying (Q1), that we call Cartan pairs. For such pairs, and the corresponding Cartan subspaces, we prove some useful results that are classical for symmetric pairs. Thus we extend a part of the previous research on Cartan subspaces done by Dixmier, Lepowsky and McCollum.

Pair of Lie algebras; semisimple Lie algebra; reductive subalgebra; Cartan subalgebra; Cartan subspace; Cartan pair

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