Embedding of Markov matrices for $d\leqslant 4$

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 $d=4$.