In general, let $A$ be a Banach Algebra, we have that $\exp(a+b) = \exp(a) + \exp(b)$ if $a$ and $b$ commute. I'm interested in the case of non-commuting $a$ and $b$ at the neighborhood of $0_A$. Specifically, fix $a, b \in A$, is it true that for all $t$ sufficiently small, we have that
$$ e^{ta}e^{tb}=e^{t(a+b+\epsilon_1(t))},$$ $$e^{ta}e^{tb}e^{-t(a+b)}=e^{t^2([a,b]+\epsilon_2(t))}$$
for some error terms $\epsilon_1(t), \epsilon_2(t) \to 0 $ as $t \to 0$. The error terms $\epsilon_1(t), \epsilon_2(t)$ are dependent on the choice of $a, b$, and $[a,b]$ is lie bracket $ab - ba$. The equation seems correct by looking the first and second order derivative with $t$, but I'm not sure if it is valid.