This paper extends the inverse substructuring (IS) approach into the state-space domain and presents a novel state-space substructuring (SSS) technique that embeds the dynamics of connecting elements (CEs) in the Lagrange Multiplier State-Space Substructuring (LM-SSS) formulation via compatibility relaxation. This coupling approach makes it possible to incorporate into LM-SSS connecting elements that are suitable for being characterized by inverse substructuring (e.g. rubber mounts) by simply using information from one of its off diagonal apparent mass terms. Therefore, the information obtained from an in-situ experimental characterization of the CEs is enough to include them into the coupling formulation. Moreover, LM-SSS with compatibility relaxation makes it possible to couple an unlimited number of components and CEs, requiring only one matrix inversion to compute the coupled state-space model (SSM). Two post-processing procedures to enable the computation of minimal-order coupled models by using this approach are also presented. Numerical and experimental substructuring applications are exploited to prove the validity of the proposed state-space realization of IS and the inclusion of CEs into the LM-SSS formulation via compatibility relaxation. It is found that the IS approach can be accurately applied on state-space models representative of components linked by CEs to identify models representative of the diagonal apparent mass terms of the CEs, provided that the CEs can be accurately characterized by the underlying assumptions of IS. In this way, state-space models representative of experimentally characterized CEs can be found without performing decoupling operations. Hence, these models are not contaminated with spurious states. Furthermore, it was found that the developed coupling approach is reliable, when the dynamics of the CEs can be accurately characterized by IS, thus making it possible to compute reliable coupled models that are not composed by spurious states.