This paper presents a sufficient condition for two vector fields $X$ and $Y$ to have the squares noncommutative, i.e. $[X^2, Y^2] \not= 0$. We prove that if the vector fields $X$, $Y$ span a nilpotent distribution with nilpotence class 2, then the squares of the vector fields do not commute.
"On a Class of Nilpotent Distributions." Taiwanese J. Math. 15 (2) 875 - 881, 2011. https://doi.org/10.11650/twjm/1500406239