Commutativity of flows

Theorem: Let $[X,Y]=0$ if and only if the flows of $X$ and $Y$ commute locally. Formally, this is expressed as $$ (\Phi^Y_t \Phi^X_s)(x) = (\Phi^X_s \Phi^Y_t)(x) $$ for all $x\in M$ and for sufficiently small $s$ and $t$ (for $s,t<{\delta }_x$ with a fixed ${\delta }_x>0$).$◼$

Proof
TFG Adrián Ruíz, teorema 4.2.
or
Thereom 7.12 Boothby$◼$

Observe that it is a local property (or if the vector fields are complete). For a counterexample see this question.
Importantly: It is true when the vector fields generate global flows, see @lee2013smooth in the second edition it's on page 233, theorem 9.44. In the first edition there's an incorrect statement of the theorem on page 469, proposition 18.5, which is corrected in the list of errata on Lee's website.