This manuscript details the SIMBa toolbox (System Identification Methods leveraging Backpropagation), which uses well-established Machine Learning tools for discrete-time linear multi-step-ahead state-space System Identification (SI). Backed up by novel linear-matrix-inequality-based free parametrizations of Schur matrices to guarantee the stability of the identified model by design, SIMBa allows for seamless integration of prior system knowledge. In particular, it can simultaneously enforce desired system properties - such as sparsity patterns - and stability on the model, solving an open SI problem. 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, and sometimes significantly, even while 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 on https://github.com/Cemempamoi/simba.