This manuscript details and extends the system identification methods leveraging the backpropagation (SIMBa) toolbox presented in previous work, which uses well-established machine learning tools for discrete-time linear multistep-ahead state-space system identification (SI). SIMBa leverages linear-matrix-inequality-based free parameterizations of Schur matrices to guarantee the stability of the identified model by design. In this article, backed up by novel free parameterizations of Schur matrices, we extend the toolbox to show how SIMBa can incorporate known sparsity patterns or true values of the state-space matrices to identify without jeopardizing stability. We extensively investigate SIMBa’s behavior when identifying diverse systems with various properties from both simulated and real-world data. Overall, we find it consistently outperforms traditional stable subspace identification methods (SIMs), and sometimes significantly, especially when enforcing desired model properties. These results hint at the potential of SIMBa to pave the way for generic structured nonlinear SI. The toolbox is open-sourced at https://github.com/Cemempamoi/simba.