Abstract Let $A, B$ be two square complex matrices of the same dimension $n\leq 3$ . We show that the following conditions are equivalent. (i) There exists a finite subset $U\subset { \mathbb{N} }_{\geq 2} $ such that for every $t\in \mathbb{N} \setminus U$ , $\exp (tA+ B)= \exp (tA)\exp (B)= \exp (B)\exp (tA)$ . (ii) The pair $(A, B)$ has property L of Motzkin and Taussky and $\exp (A+ B)= \exp (A)\exp (B)= \exp (B)\exp (A)$ . We also characterise the pairs of real matrices $(A, B)$ of dimension three, that satisfy the previous conditions.