The <Lie bracket of vector fields> is the commutator of the corresponding derivations:
$$
[X,Y]f=X(Yf)-Y(Xf).
$$
For $X=X^i\partial_i$ and $Y=Y^i\partial_i$,
$$
\boxed{[X,Y]=\left(X^i\partial_iY^k-Y^i\partial_iX^k\right)\partial_k.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 2 / e / Solution

If $X$ and $Y$ preserve $I(N)$, then their commutator preserves it because
$$
[X,Y]f=X(Yf)-Y(Xf)\in I(N)
$$
for every $f\in I(N)$. Thus vector fields tangent to an embedded submanifold are closed under the <Lie bracket of vector fields>, and the restricted bracket depends only on their restrictions to the submanifold.

Tangency under the Lie bracket

{c}
{title2=$[X,Y]$}
{wiki}

The Lie bracket is the vector field defined as a commutator of derivations,
$$
[X,Y]f=X(Yf)-Y(Xf).
$$
In coordinates, if $X=X^i\partial_i$ and $Y=Y^i\partial_i$, then $[X,Y]^k=X^i\partial_iY^k-Y^i\partial_iX^k$.


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