The embedding problem of Markov matrices in Markov semigroups is a classic problem that regained a lot of impetus and activities through recent needs in phylogeny and population genetics. Here, we give an account for dimensions \(d\leqslant 4\) , including a complete and simplified treatment of the case \(d=3\) , and derive the results in a systematic fashion, with an eye on the potential applications. Further, we reconsider the setup of the corresponding problem for time-inhomogeneous Markov chains, which is needed for real-world applications because transition rates need not be constant over time. Additional cases of this more general embedding occur for any \(d\geqslant 3\) . We review the known case of \(d=3\) and describe the setting for future work on \(d=4\) .
