  Deciding whether a Markov matrix is embeddable (i.e. can be written as the
exponential of a rate matrix) is an open problem even for $4\times 4$ matrices.
We study the embedding problem and rate identifiability for the K80 model of
nucleotide substitution. For these $4\times 4$ matrices, we fully characterize
the set of embeddable K80 Markov matrices and the set of embeddable matrices
for which rates are identifiable. In particular, we describe an open subset of
embeddable matrices with non-identifiable rates. This set contains matrices
with positive eigenvalues and also diagonal largest in column matrices, which
might lead to consequences in parameter estimation in phylogenetics. Finally,
we compute the relative volumes of embeddable K80 matrices and of embeddable
matrices with identifiable rates. This study concludes the embedding problem
for the more general model K81 and its submodels, which had been initiated by
the last two authors in a separate work.
