Let $M$ be a $n$-dimensional smooth manifold, and $X$ a (globally defined) smooth vector field on $M$. Is it always possible to write $X$ as a Lie bracket $[X_1,X_2]$ of two other globally defined smooth vector fields?
Clearly this is possible locally: Choosing a coordinate patch $O$ diffeomorphic to $[0,1]^n$, we can write $X = \sum_{j=1}^nX^j \frac{\partial}{\partial x^j}$ and solve $[\frac{\partial}{\partial x^1}, Y] = \sum_{j=1}^n (\frac{\partial Y^j}{\partial x^1}) \frac{\partial}{\partial x^j} = \sum_{j=1}^n X^j \frac{\partial}{\partial x^j}$ to obtain $X$ as a Lie bracket $[\frac{\partial}{\partial x^1}, Y]$. But, globally, a nowhere vanishing vector field to play the role of $\frac{\partial}{\partial x^1}$ in the previous calculation need not exist, and even if it exists, the resulting ODE for the coefficients of $Y$ might run into topological trouble.
To give another simple example, the question has a positive answer for the unit circle $\mathbb{S}^1$ as well: Every vector field $X = f(\theta)\frac{\partial}{\partial \theta}$ can be written as a Lie bracket by the following trick: Let $c = \int_0^{2\pi} f(\theta)d\theta$, and $\gamma(\theta) = \frac{2\pi}{c} \int_0^\theta f(\theta)d\theta$. Then $\sin\circ \gamma$ and $\cos\circ \gamma$ are smooth functions on $\mathbb{S}^1$, and $$\left[\frac{c}{2\pi}\sin(\gamma(\theta))\frac{\partial}{\partial \theta}, \cos(\gamma(\theta))\frac{\partial}{\partial \theta}\right] = \frac{c}{2\pi}\left(\sin(\gamma(\theta))^2+\cos(\gamma(\theta))^2\right) \gamma'(\theta) \frac{\partial}{\partial \theta} = f(\theta)\frac{\partial}{\partial \theta}.$$ It is not possible to write $\frac{\partial}{\partial \theta}$ as $[\frac{\partial}{\partial \theta},X]$ for any vector field $X$, on the other hand.