= Complete vector fields need not be closed under addition or Lie brackets
On $\mathbb R^2$, the <vector fields> $X=y\partial_x$ and $Y=(x^2/2)\partial_y$ are complete, since their <local flows> are $(x+ty,y)$ and $(x,y+tx^2/2)$. Their <Lie bracket of vector fields> is $-(x^2/2)\partial_x+xy\partial_y$. Its <integral curve of a vector field> starting at $(2,0)$ is $(2/(1+t),0)$, which cannot continue through $t=-1$. The sum has an <integral curve of a vector field>
$$
\left(\left(1-\frac{t}{2\sqrt3}\right)^{-2},\frac1{\sqrt3}\left(1-\frac{t}{2\sqrt3}\right)^{-3}\right)
$$
that cannot continue through $t=2\sqrt3$. Thus neither operation preserves <complete vector fields> on a general noncompact <smooth manifold>. This contrasts with a <compact manifold> without boundary, where every smooth <vector field> is complete.
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