A question about tangent subbundle Let $x$ be a manifold, $E$ is a subbundle of  $TX$ , my question is :
Can you give example such that  vector fields $\xi ,\eta$ lie in $E$,but bracket $[\xi ,\eta]$ does not lie in $E$ in some point of $x$.
 A: Consider the tangent space to $R^3$, which we can write as $R^3 \times R^3$. At each point $(x, y, z)$, consider the plane orthogonal to $(-y, x, 1)$. These planes form a subbundle. 
Now look at two vector fields, one radial from the origin, 
$$
F(x, y, z) = \begin{bmatrix}x\\ y\\ 0\end{bmatrix}
$$
the other "circumferential", i.e., 
$$
G(x, y) = \begin{bmatrix}-y\\ x\\ -(x^2+y^2)\end{bmatrix}.
$$
(I had originally written
$$
G(x, y, z) = [-y, x, 0].
$$
which doesn't even lie in the specified plane, and the OP complained that the bracket, for that $G$, turns out to be zero. D'oh!) 
As the OP noted in comments, in coordinates, the bracket is just 
\begin{align}
[F, G] 
&= G' F - F' G\\
&= \begin{bmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
-2x & -2y & 0
\end{bmatrix}
\begin{bmatrix}
x\\ y\\ 0\end{bmatrix}
-\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
-y\\ x\\ -(x^2+y^2)\end{bmatrix} \\
&= \begin{bmatrix}
-y\\
x\\
-2(x^2 + y^2)
\end{bmatrix} -
\begin{bmatrix}
-y\\ x\\ 0
\end{bmatrix}\\ 
&= \begin{bmatrix}
0\\
0\\
-2(x^2 + y^2)
\end{bmatrix}
\end{align}
That bracket is nonzero, and points in the $z$-direction, i.e., a direction not in the plane (except at the origin, of course). 
