[math-fun] exp(A)exp(B) = exp(A+B) = exp(B)exp(A)

It's well known and easy to prove that if NxN matrices A and B commute, then (*) exp(A)exp(B) = exp(A+B) = exp(B)exp(A) . But AB = BA is not a necessary condition for (*), as googling will readily reveal. Does anyone know necessary and sufficient conditions on A and B for (*) to hold? If not, how about a large class of A and B for which (*) holds despite AB and BA being unequal ? I think such examples of complex matrices are easier to come by than of real ones, so I'm particularly interested in the case where A and B are real. --Dan