Motivated by the way two special domains, namely the symmetrized bidisc and the tetrablock, could be defined as the images of 2-proper holomorphic images of classical Car-tan domains, we present a general approach to study 2-proper holomorphic images of bounded symmetric domains. We show some special properties of 2-proper holomorphic maps (such as the construction of some involutive automorphisms, etc.) and enlist possible domains (up to biholomorphisms) which arise as 2-proper holomorphic images of bounded symmetric domains. This leads us to a consideration of a new family of domains Ln for n >= 2. Let Ln be an irreducible classical Cartan domain of type IV (Lie ball) of dimension n, and Lambda(n) : L-n -> Lambda(n)(L-n) := L(n )be the natural proper holomorphic mapping of multiplicity 2. It turns out that L-2 and L-3 are biholomorphic to the symmetrized bidisc and the tetrablock, respectively. In this article, we study function geometric properties of the family {L-n : n >= 2} in a unified manner, and thus extend results of many earlier papers on analogous properties of the symmetrized bidisc and the tetrablock. We show that L-n cannot be exhausted by domains biholomorhic to some convex domains. Any proper holomorphic self-mapping of L-n is an automorphism for n >= 3. Moreover, the automorphism group Aut(L-n) is isomorphic to Aut(Ln-1), and L-n is inhomogeneous for n >= 2. Additionally, we prove that L-n is not a Lu Qi-Keng domain for n >= 3.