Two commuting real matrices $A$ and $B$ have commuting exponentials $\exp A$ and $\exp B$, a fact observed for instance in linear algebra or differential equations courses. The converse implication is false. A clarification of this phenomenon is proposed that makes use of the theory of the exponential function $\exp\colon{\fam\euffam g}\to G$ of a real Lie group $G$ and its singularities. In Section 1, a catalog of low-dimensional examples illustrates various ways that, for two elements $X, Y\in{\fam\euffam g}$, the commuting of $\exp X$ and $\exp Y$ in $G$ may fail to entail the commuting of $X$ and $Y$ in ${\fam\euffam g}$. In Section 2, consequences of the relation $[\exp X,\exp Y]={\bf 1}$ are inspected, whereby certain regularity assumptions on $X$ and $Y$ are made. A regular element $Y$ of the Lie algebra ${\fam\euffam g}$ determines a Cartan subalgebra ${\fam\euffam h}={\fam\euffam g}^0(Y)$ of ${\fam\euffam g}$ and a certain subgroup ${\cal W}_Y$ of the (finite!) Weyl group of ${\fam\euffam g}$ with respect to the Cartan subalgebra ${\fam\euffam h}$. If, additionally, the exponential function is regular at $X$ and at $Y$, then the ordered pair $(X,Y)$ is said to be in general position. If $(X,Y)$ is in general position, then the relation $[\exp X,\exp Y]={\bf 1}$ in $G$ permits the definition of a certain element $w(X,Y)\in{\cal W}_Y$. Let ${\fam\euffam z}({\fam\euffam g})$ denote the center of ${\fam\euffam g}$. It is shown that, if $\exp X$ and $\exp Y$ commute in $G$ for $(X,Y)$ in general position, then $[X,Y]\in{\fam\euffam z}({\fam\euffam g})\cap[{\fam\euffam h},{\fam\euffam h}]$ iff $w(X,Y)={\bf 1}$. Write $H\defi\exp{\fam\euffam h}$, and let $Z(G)$ denote the center of $G$. If the identity component of $Z(G)\cap[H,H]$ is simply connected, and if $\exp X$ and $\exp Y$ commute for $(X,Y)$ in general position, then $[X,Y]=0$ iff $w(X,Y)={\bf 1}$. If $G$ is simply connected compact, then $[\exp X,\exp Y]={\bf 1}$ and $[X,Y]=0$ are equivalent for all pairs $(X,Y)$ in general position. In ${\mathop{\rm SO}\nolimits}(3)$ this is not the case; here $|{\cal W}_Y|=2$. In Section 3, examples show that the validity of the equation $\exp X\exp Y=\exp\!(X+Y)$ has no implications whatsoever in the direction of the commuting of $\exp X$ and $\exp Y$. Finally, in Section 4, it is shown that, for a simply connected Lie group $G$, the commuting of $X, Y\in{\fam\euffam g}$ and that of $\exp X,\exp Y\in G$ are equivalent properties for all$X$ and $Y$ if and only if the exponential function is injective. This class of Lie groups was characterized in terms of other properties by Dixmier and by Saito, independently, in 1957.