On commuting exponentials in low dimensions

We deal with square matrices A, B of dimension d = 2 or 3, over the complex field, such that AB ≠ BA. We introduce the relations Gt:exp(tA+B)=exp(tA)exp(B) and . In dimension 2, we characterize the (A, B) couples satisfying Gt for any t∈N. In dimension 2 or 3, we show that if is satisfied for any t∈N, then A and B are simultaneously triangularizable. In this manner we do not need the 2iπ-congruence-free postulate anymore, which has been supposed by researchers since 1954.