In 2008, we obtained an intrinsic characterization of the unit polydisc Δn in Cn from the viewpoint of the holomorphic automorphism group. In connection with this, A. V. Isaev investigated the structure of a complex manifold M with the property that every isotropy subgroup of the holomorphic automorhism group of M is compact, and obtained the same characterization of Δn as ours among the class of all such manifolds. In this paper, we establish some extensions of these results. In particular, Isaev's characterization of the unit polydisc Δn is extended to that of any bounded symmetric domain in Cn.