Supersymmetry algebras can be used to obtain algebraic expressions for constant Yang-Baxter solutions, also known as braid group generators. This was done for non-invertible braid operators in Maity et al (2025 J. High Energy Phys 5 1-31) In this work we extend this construction for the invertible ones. The resulting expressions are then shown to obey relations analogous to those satisfied by quotients of braid groups. Examples of the latter include the Iwahori-Hecke algebra and the Birman-Murakami-Wenzl algebra. As a result, we can Baxterize the constant Yang-Baxter solutions to yield spectral parameter dependent R-matrices. The regularity of these R-matrices depends on the representation of SUSY generators. In some cases they are regular in the usual sense and in the remaining they are 'almost' regular. In the latter case they are also non-invertible. Nevertheless, we show that they can still help us construct integrable models in all dimensions of the local Hilbert space. These models can be described by Hamiltonian densities that are either local or non-local, depending on the representation chosen for the SUSY generators. We demonstrate this for all constant 4 & times;4 invertible Yang-Baxter solutions. Apart from reproducing nearest-neighbor interaction spin 12 systems, we also find their higher spin analogs due to the algebraic [representation independent] approach.