FMMTL: FMM template library a generalized framework for kernel matrices. In response to two decades of development in structured dense matrix algorithms and a vast number of research codes, we present designs and progress towards a codebase that is abstracted over the primary domains of research. In the domain of mathematics, this includes the development of interaction kernels and their low-rank expansions. In the domain of high performance computing, this includes the optimized construction, traversal, and scheduling algorithms for the appropriate operations. We present a versatile system that can encompass the design decisions made over a decade of research while providing an abstracted, intuitive, and usable front-end that can integrated into existing linear algebra libraries.