We study a class of pairs of Lie algebras $(\g, \g_1)$ that we call Cartan pairs ; here $\g$ is semisimple and $\g_1$ is a reductive in $\g$ subalgebra. For these pairs, which generalize symmetric ones, we have standardly defined Cartan subspaces, and consequently the set of restricted roots $\Sigma(\g, \a)$. We prove that there are infinitely many interesting nonsymmetric Cartan pairs. Next we prove that every pair of the well known Brylinski-Kostant list of shared orbit pairs is a Cartan pair. As a continuation of the previous research we obtained some further useful and clarifying results and examples related to Cartan pairs and Cartan subspaces.