  We prove that the probability substitution matrices obtained from a
continuous-time Markov chain form a multiplicatively closed set if and only if
the rate matrices associated to the chain form a linear space spanning a Lie
algebra. The key original contribution we make is to overcome an obstruction,
due to the presence of inequalities that are unavoidable in the probabilistic
application, that prevents free manipulation of terms in the
Baker-Campbell-Haursdorff formula.
