  We provide a characterisation of the continuous-time Markov models where the
Markov matrices from the model can be parameterised directly in terms of the
associated rate matrices (generators). That is, each Markov matrix can be
expressed as the sum of the identity matrix and a rate matrix from the model.
We show that the existence of an underlying Jordan algebra provides a
sufficient condition, which becomes necessary for (so-called) linear models. We
connect this property to the well-known uniformization procedure for
continuous-time Markov chains by demonstrating that the property is equivalent
to all Markov matrices from the model taking the same form as the corresponding
discrete time Markov matrices in the uniformized process. We apply our results
to analyse two model hierarchies practically important to phylogenetic
inference, obtained by assuming (i) time-reversibility and (ii) permutation
symmetry, respectively.
